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Grade 10 Mathematics Teachers’ Book

INNODEMS

Subsection 1.3.2 Quadratic Identities

In algebra, an identity is an equation that is true for all values of the variable.
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A quadratic identity is an identity in which the highest power of the variable is \(2\text{.}\)
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For example: \((x+1)^2 = x^2 + 2x + 1\)
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Both sides contain the variable \(x\text{,}\) and both sides give the same result for any value of \(x\text{.}\) Therefore, this equation is an identity.
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In this section, we will learn important quadratic identities that help us expand and factorise algebraic expressions.
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  • Perfect Square identities
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  • Difference of Squares identities
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Note 1.3.21.

Important quadratic identities are useful because they make algebraic calculations faster and easier. You should learn the main ones and practise using them.
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Subsubsection 1.3.2.1 Perfect Squares

Learner Experience 1.3.6.

Study the diagram below carefully. Use what you know about area to investigate what is happening.
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A square with side length (a+b) divided into four colored regions representing a^2, 2ab, and b^2
Figure 1.3.23. Geometric Decomposition of \((a+b)^2\)
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  1. Find the areas of the smaller regions
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    1. Identify the side lengths of each colored region; the blue square, the two green rectangles, and the red square.
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    2. Add all these smaller areas together.
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    Write your total area as a simplified algebraic expression.
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  2. Find the area of the large square
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    1. What is the total length of one side of the large outer square?
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    2. Using the formula for the area of a square, calculate the area of the whole square directly.
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    3. Express your answer in algebraic form.
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  3. Compare your results
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    1. Compare the total area you found by adding the smaller parts with the area you found using the side length of the large square.
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    2. Are the two expressions the same?
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    3. What conclusion can you make about \((a+b)^2\text{?}\)
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  4. How does breaking a square into smaller regions help explain why \((a+b)^2\) expands the way it does?
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  5. What does each smaller region represent in the expansion?
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Key Takeaway 1.3.24.

An integer that can be written as the product of two equal integers is called a perfect square. For example, \(16\) is a perfect square because \(16 = 4 \times 4\text{,}\) and \(25\) is a perfect square because \(25 = 5 \times 5\text{.}\)
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In algebra, a perfect square identity describes what happens when a binomial
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A binomial is an algebraic expression made up of exactly two terms joined by a plus (\(+\)) or minus (\(-\)) sign.
is multiplied by itself. The result is a quadratic expression.
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The two main perfect square identities are:
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  • \(\displaystyle (a+b)^2 = a^2 + 2ab + b^2\)
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  • \(\displaystyle (a-b)^2 = a^2 - 2ab + b^2\)
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These identities are important because they help us expand expressions and recognize special patterns when factoring quadratic expressions.
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Example 1.3.25.

Expand and simplify: \((x + 5)^2\text{.}\)
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Solution.
Use the perfect square identity \((a+b)^2 = a^2 + 2ab + b^2\text{.}\)
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Here, \(a = x\) and \(b = 5\text{.}\)
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\((x+5)^2 = x^2 + 2(x)(5) + 5^2 = x^2 + 10x + 25.\)
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Example 1.3.26.

Expand and simplify: \((2y - 3)^2\text{.}\)
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Solution.
Use the perfect square identity \((a-b)^2 = a^2 - 2ab + b^2\text{.}\)
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Here, \(a = 2y\) and \(b = 3\text{.}\)
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\((2y - 3)^2 = (2y)^2 - 2(2y)(3) + 3^2 = 4y^2 - 12y + 9.\)
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Example 1.3.27.

Show that \(9x^2 + 24x + 16\) is a perfect square and express it in factorized form.
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Solution.
Compare the expression with the identity \((a+b)^2 = a^2 + 2ab + b^2\text{.}\)
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\(9x^2 = (3x)^2, \quad 16 = 4^2, \quad \text{and} \quad 2(3x)(4) = 24x.\)
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Therefore, \(9x^2 + 24x + 16 = (3x + 4)^2.\)
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Checkpoint 1.3.28.

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Checkpoint 1.3.29.

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Exercises Exercises

1.
Expand and simplify the following expressions:
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  1. \(\displaystyle (x + 7)^2\)
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  2. \(\displaystyle (3a - 2)^2\)
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  3. \(\displaystyle (2m + 5n)^2\)
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Answer.
  1. \(\displaystyle x^2 + 14x + 49\)
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  2. \(\displaystyle 9a^2 - 12a + 4\)
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  3. \(\displaystyle 4m^2 + 20mn + 25n^2\)
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2.
Show that \(x^2 - 10x + 25\) is a perfect square and write it in factorized form.
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Answer.
\(x^2 - 10x + 25 = (x - 5)^2\)
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3.
Factorize: \(16y^2 + 40y + 25\text{.}\)
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Answer.
\(16y^2 + 40y + 25 = (4y + 5)^2\)
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4.
Determine whether \(4x^2 + 12x + 7\) is a perfect square. Give a reason for your answer.
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Answer.
It is not a perfect square because \(4x^2 = (2x)^2\) and \(7\) is not a perfect square, and the middle term \(12x\) does not match \(2(2x)(\sqrt{7})\text{.}\) Therefore, it cannot be written in the form \((a+b)^2\text{.}\)
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Subsubsection 1.3.2.2 Difference of Squares

Learner Experience 1.3.7.

Work in pairs or groups
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What you require: Graph paper, a pair of scissors, a ruler, and a pencil.
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(a)
On your graph paper, draw a large square with side length \(a\) (for example, let \(a = 10\) units). Calculate its area (\(A = a^2\)).
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(b)
In the top-left corner of your large square, draw a smaller square with side length \(b\) (for example, let \(b = 4\) units).
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(c)
Shade this small square. Its area is \(b^2\text{.}\)
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(d)
The unshaded region represents the area of the large square minus the small square. Write down the expression for this area (\(a^2 - b^2\)).
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(e)
Draw a horizontal line extending from the bottom of the small square to the right edge of the large square. This divides the unshaded region into two rectangles.
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(f)
Cut out the L-shaped unshaded region from your paper. Then, cut along the horizontal line you drew to separate it into two rectangles.
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(g)
Rearrange the pieces: Take the smaller rectangle and place it next to the larger rectangle to form one long, single rectangle.
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(h)
Measure the length and width of this new large rectangle in terms of \(a\) and \(b\text{:}\)
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(i)
Calculate the area of this new rectangle using the formula \(\text{Length} \times \text{Width}\text{.}\)
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(j)
Compare the area from step (d) with the area from step (i). Since the amount of paper has not changed, conclude the relationship between the two expressions.
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Figure 1.3.30.
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Key Takeaway 1.3.31.

The Difference of Squares is a quadratic identity used when two perfect squares are subtracted. It states that:
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\begin{align*} a^2 - b^2 \amp = (a - b)(a + b) \end{align*}
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This means that the difference of two perfect squares can be factorised into two binomials: one with subtraction and one with addition.
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In the expression \(a^2 - b^2\text{,}\) the terms \(a\) and \(b\) are the square roots of the two perfect squares. Therefore, whenever you see two perfect squares being subtracted, you can factorise them as \((a - b)(a + b)\text{.}\)
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Example 1.3.32.

Factorise each of the following expressions completely:
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  1. \(\displaystyle x^2 - 49\)
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  2. \(\displaystyle 16y^2 - 25\)
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  3. \(\displaystyle 9a^2 - 4b^2\)
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Hint.
Recall the Difference of Two Squares identity:
\begin{equation*} a^2 - b^2 = (a - b)(a + b) \end{equation*}
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Solution.
To use this identity, we must first express both terms as \((\text{something})^2\text{.}\)
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  1. Factorising \(x^2 - 49\text{:}\)
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    • Rewrite as squares: \(x^2 - 7^2\)
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    • Identify \(a\) and \(b\text{:}\) Here, \(a = x\) and \(b = 7\text{.}\)
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    • Apply Identity: \((x - 7)(x + 7)\)
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  2. Factorising \(16y^2 - 25\text{:}\)
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    • Rewrite as squares: \((4y)^2 - 5^2\)
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    • Identify \(a\) and \(b\text{:}\) Here, \(a = 4y\) and \(b = 5\text{.}\)
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    • Apply Identity: \((4y - 5)(4y + 5)\)
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  3. Factorising \(9a^2 - 4b^2\text{:}\)
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    • Rewrite as squares: \((3a)^2 - (2b)^2\)
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    • Identify \(a\) and \(b\text{:}\) Here, \(a = 3a\) and \(b = 2b\text{.}\)
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    • Apply Identity: \((3a - 2b)(3a + 2b)\)
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Checkpoint 1.3.33.

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Checkpoint 1.3.34.

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Exercises Exercises

1.
Factorise completely the following expressions:
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  1. \(\displaystyle 25x^2 - 81\)
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  2. \(\displaystyle 49a^2b^2 - 16\)
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  3. \(\displaystyle 8x^2 - 18\)
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  4. \(\displaystyle m^4 - 81\)
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  5. \(\displaystyle 36x^2y^2 - 121z^2\)
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Answer.
  1. \(\displaystyle (5x - 9)(5x + 9)\)
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  2. \(\displaystyle (7ab - 4)(7ab + 4)\)
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  3. \(\displaystyle 8x^2 - 18 = 2(4x^2 - 9) = 2(2x - 3)(2x + 3)\)
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  4. \(\displaystyle m^4 - 81 = (m^2 - 9)(m^2 + 9) = (m - 3)(m + 3)(m^2 + 9)\)
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  5. \(\displaystyle (6xy - 11z)(6xy + 11z)\)
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Subsubsection 1.3.2.3 Quadratic Identities in Numerical Cases

Learner Experience 1.3.8.

Work in groups Scenario: Imagine you are planning a new garden in your backyard. You want to create a rectangular vegetable patch that has an area of \(48\) square feet. You decide to set the length of the garden to be twice the width.
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(a)
Let \(w\) represent the width of the garden. If the length \(l = 2w\text{,}\) write an equation for the area \(A = l \times w\text{.}\)
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(b)
Subsitute \(l = 2w\) into the area equation. Show that it simplifies to \(2w^2 = 48\)
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(c)
Solve for \(w\) to find the width, then calculate the length.
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(d)
Verify your answer by checking that \(l \times w = 48\text{.}\)
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(e)
Discussion: Suppose you wanted a square garden instead, with a side length of \((x + 3)\) metres. Discuss with your group how you would calculate the area without just multiplying the numbers directly.
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Key Takeaway 1.3.36.

Quadratic Identities are standard patterns that help us expand expressions or calculate large numbers mentally without using long multiplication.
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There are three main identities we use:
  1. Perfect Square (Sum):
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    \((a+b)^2 = a^2 + 2ab + b^2\)
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  2. Perfect Square (Difference):
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    \((a-b)^2 = a^2 - 2ab + b^2\)
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  3. Difference of Two Squares:
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    \((a+b)(a-b) = a^2 - b^2\)
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We can use these identities to solve numerical problems (like finding \(103^2\)) or to find areas in real-world problems.
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Example 1.3.37.

Use a quadratic identity to calculate \(103^2\) without using a calculator.
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Solution.
We can rewrite \(103\) as \((100 + 3)\text{.}\) Using the identity \((a+b)^2 = a^2 + 2ab + b^2\text{:}\)
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\(\begin{aligned} 103^2 \amp = (100 + 3)^2 \\ \amp = 100^2 + 2(100)(3) + 3^2 \\ \amp = 10,000 + 600 + 9 \\ \amp = \mathbf{10,609} \end{aligned}\)
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Example 1.3.38.

A square garden has a side length of \((x+3)\) metres. Find the area using a quadratic identity. If \(x=5\text{,}\) what is the area in square metres?
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Solution.
The area of a square is given by \((\text{side})^2\text{.}\) Using the identity \((a+b)^2 = a^2 + 2ab + b^2\text{:}\)
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\(\begin{aligned} \text{Area} \amp = (x+3)^2 \\ \amp = x^2 + 2(x)(3) + 3^2 \\ \amp = x^2 + 6x + 9 \end{aligned}\)
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Now, substitute \(x=5\text{:}\)
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\(\begin{aligned} \text{Area} \amp = 5^2 + 6(5) + 9 \\ \amp = 25 + 30 + 9 \\ \amp = \mathbf{64 \text{ m}^2} \end{aligned}\)
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Checkpoint 1.3.39.

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Checkpoint 1.3.40.

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Checkpoint 1.3.41.

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Exercises Exercises

1.
Use a quadratic identity to evaluate \(202^2\text{.}\)
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Answer.
\(40\,804\)
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2.
Calculate \(63 \times 57\) using the difference of squares identity.
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Answer.
\(3\,591\)
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3.
A rectangular field has dimensions \((x+5)\) metres by \((x-5)\) metres. Express the area using a quadratic identity. If \(x=12\text{,}\) find the area.
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Answer.
\(119\) m\(^2\)
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4.
A square playground has a side length of \((y+7)\) metres. Write the area as a quadratic expression and find the area when \(y=3\text{.}\)
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Answer.
\(100\) m\(^2\)
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Exercises 1.3.2.4 Exercises

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Use the quadratic identities to write down the expansions of each of the following expressions:
  1. \(\displaystyle (4x + 5)^2\)
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  2. \(\displaystyle \left( \frac{1}{x} + \frac{1}{y} \right) \left( \frac{1}{x} - \frac{1}{y} \right)\)
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  3. \(\displaystyle (8 - x)^2\)
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  4. \(\displaystyle (x - 7)^2\)
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  5. \(\displaystyle \left( x + \frac{1}{2} \right)^2\)
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  6. \(\displaystyle \left( \frac{1}{4} - \frac{3}{4}b \right)^2\)
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  7. \(\displaystyle (x + 2)^2\)
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  8. \(\displaystyle (x + 5)^2\)
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Answer.
  1. Using the perfect squares identity:
    \begin{align*} (4x + 5)^2 \amp = (4x)^2 + 2(4x)(5) + 5^2\\ \amp = 16x^2 + 40x + 25 \end{align*}
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  2. Using the difference of squares identity:
    \begin{align*} \left( \frac{1}{x} + \frac{1}{y} \right) \left( \frac{1}{x} - \frac{1}{y} \right) \amp = \left( \frac{1}{x} \right)^2 - \left( \frac{1}{y} \right)^2\\ \amp = \frac{1}{x^2} - \frac{1}{y^2} \end{align*}
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  3. Using the perfect squares identity:
    \begin{align*} (8 - x)^2 \amp = 8^2 - 2(8)(x) + x^2\\ \amp = x^2 - 16x + 64 \end{align*}
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  4. Using the perfect squares identity:
    \begin{align*} (x - 7)^2 \amp = x^2 - 2(x)(7) + 7^2\\ \amp = x^2 - 14x + 49 \end{align*}
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  5. Using the perfect squares identity:
    \begin{align*} \left( x + \frac{1}{2} \right)^2 \amp = x^2 + 2\left(x\right)\left(\frac{1}{2}\right) + \left(\frac{1}{2}\right)^2\\ \amp = x^2 + x + \frac{1}{4} \end{align*}
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  6. Using the perfect squares identity:
    \begin{align*} \left( \frac{1}{4} - \frac{3}{4}b \right)^2 \amp = \left( \frac{1}{4} \right)^2 - 2\left( \frac{1}{4} \right)\left( \frac{3}{4}b \right) + \left( \frac{3}{4}b \right)^2\\ \amp = \frac{1}{16} - \frac{3}{8}b + \frac{9}{16}b^2 \end{align*}
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  7. Using the perfect squares identity:
    \begin{align*} (x + 2)^2 \amp = x^2 + 2(x)(2) + 2^2\\ \amp = x^2 + 4x + 4 \end{align*}
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  8. Using the perfect squares identity:
    \begin{align*} (x + 5)^2 \amp = x^2 + 2(x)(5) + 5^2\\ \amp = x^2 + 10x + 25 \end{align*}
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