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Grade 10 Mathematics Learners’ Book

INNODEMS

Subsection 1.3.5 Solutions Of Quadratic Equations By Factorisations.

Activity 1.3.11.

Work in Groups.
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Find the roots of the equation using factorization.
  1. \(x² - 5x + 6 = 0\text{.}\)
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  2. \(x² + 7x + 10 = 0\text{.}\)
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After solving the equations, discuss the following:
  • What steps did you follow in factorizing the quadratic equation?
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  • How do the roots relate to the factors of the quadratic equation?
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  • Did any group use a different method to factorize the equation? Compare approaches.
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Finally, each group will share their solved quadratic equations and the methods they used infront of the classroom.
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In our previous sections we have discussed the form of a quadratic expression, identities and quadratic equations; how to form quadratic equations and differentiate between an expression and equation
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In this section we are going to learn about solving the factored equation, we are going to continue from the point where we form factored equation
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Numbers that satisfy an equation (its solutions) are called the roots of the equation.
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Once you have factored the quadratic into the form;
\begin{align*} (x+p)(x+q) \amp = 0 \end{align*}
Set each factor equal to zero and solve for \(x\text{:}\)
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\begin{align*} (x + p) \amp = 0 \end{align*}
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\begin{align*} \text{or} \amp \end{align*}
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\begin{align*} (x + q) \amp = 0 \end{align*}
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Solving these will give the two solutions for \(x\text{.}\)
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Example 1.3.17.

Solve
\begin{align} x^2 + 5x + 6 \amp = 0 \tag{1.3.1} \end{align}
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Solution.
Look for two numbers that if multiplied by \(6\) and when added up gives \(5\)
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These numbers are \(2\) and \(3.\) So we can write the equation as:
\begin{align*} x^2 + 2x + 3x + 6 \amp = 0 \end{align*}
Finding a common factor we have:
\begin{align*} x(x + 2) + 3(x + 2) \amp = 0 \end{align*}
Grouping we have th factors of the equation as:
\begin{align*} (x +2) (x + 3) \amp = 0 \end{align*}
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Setting each factor equal to zero (0).
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\begin{equation*} x + 2 = 0 \end{equation*}
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\begin{equation*} \text{or} \end{equation*}
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\begin{equation*} x + 3 = 0 \end{equation*}
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Solving for \(x.\)
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\begin{equation*} x = -2 \end{equation*}
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\begin{equation*} \text{or} \end{equation*}
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\begin{equation*} x = -3 \end{equation*}
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Thus, the solutions to the quadratic equation \(x^+5x+6=0\) are;
\begin{equation*} x = (-2) \, \text {and} \, x = (-3). \end{equation*}
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Example 1.3.18.

Solve,
\begin{align} 6x^2 + 13x + 6 \amp = 0 \tag{1.3.2} \end{align}
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Solution.
We need 2 numbers to form factors, which when;
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These numbers are \(9\) and \(4.\) So rewrite the middle part.
\begin{align*} 6x^2 + 9x + 4x + 6 \amp = 0 \end{align*}
Grouping the factors:
\begin{align*} (6x^2 + 9x) + (4x + 6) \amp = 0 \end{align*}
Factor each group:
\begin{align*} 3x(2x + 3) + 2(2x + 3) \amp = 0 \end{align*}
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\begin{align*} (2x+3)(3x+2) \amp = 0 \end{align*}
Set each factor eaqual to zero:
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\begin{equation*} 2x + 3 = 0 \end{equation*}
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\begin{equation*} \text{or} \end{equation*}
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\begin{equation*} 3x + 2 = 0 \end{equation*}
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Solving for \(x.\)
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\begin{equation*} x = -\frac{3}{2} \end{equation*}
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\begin{equation*} \text{or} \end{equation*}
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\begin{equation*} x = -\frac{2}{3} \end{equation*}
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Thus, the solutions to \(6x^2 +13x+6=0\) are:
\begin{equation*} x = -\frac{3}{2} \, \text{and} \, x = -\frac{2}{3} \end{equation*}
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Exercises Exercises

1.

Solve the quadratic equation by factorisation.
  1. \(\displaystyle x^2 + 7x + 10 = 0\)
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  2. \(\displaystyle x^2 - 5x + 6 = 0\)
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  3. \(\displaystyle x^2 + 3x - 4 = 0\)
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  4. \(\displaystyle 2x^2 + 9x + 7 = 0\)
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  5. \(\displaystyle x^2 - 6x + 8 = 0\)
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2.

  1. A car’s speed is represented by a quadratic equation:
    \begin{align*} 4x^2 - 16x + 15 \amp = 0 \text{.} \end{align*}
    Find the possible values of x representing time.
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  2. The sum of a number and its square is \(42\text{.}\) Form a quadratic equation and solve it to find the number.
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  3. Solve the quadratic equation:
    \begin{equation*} 3x^2 - 14x + 8 = 0 \end{equation*}
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  4. A garden’s area is 56 square meters, and its length is 4 meters more than its width. Form and solve a quadratic equation to find the dimensions of the garden.
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