Question

Factor and use the zero-product property to find the roots of the following quadratic equation.
(a) $0={{x}^{2}}-7x+12$
(b) $0=6{{x}^{2}}-23x+20$
(c) $0={{x}^{2}}-9$
(d) $0={{x}^{2}}+12x+36$

Answer 1

First understand the definition of zero product property. To factorize the quadratic expression in (a) and (b) apply the middle term split method. Use the algebraic identity: - ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ to factorize the expression in (c). For (d) use the formula: - ${{a}^{2}}+2ab+{{b}^{2}}={{\left( a+b \right)}^{2}}$ for factorization.

Complete step-by-step solution:
Here, we have been provided with four quadratic equations and we are asked to factorize them and use the zero-product property to find the roots. But first we need to know about the zero-product property.
Now, in mathematics, the zero product property states that if m and n are two non – zero numbers then their product will not be zero. In other words, if $m\times n=0$ then either m = 0 and n = 0.
Now, let us come to the quadratic equations one – by – one.
(a) $0={{x}^{2}}-7x+12$
$\Rightarrow {{x}^{2}}-7x+12=0$
Using the middle term split method, we have,

& \Rightarrow {{x}^{2}}-4x-3x+12=0 \\\ & \Rightarrow \left( x-4 \right)\left( x-3 \right)=0 \\\ \end{aligned}$$ Applying the zero-product property, we get, $$\Rightarrow $$ either x = 4 or x = 3 Therefore, x = 4 or x = 3 is the solution. (b) $$0=6{{x}^{2}}-23x+20$$ $$\Rightarrow 6{{x}^{2}}-23x+20=0$$ Using the middle term split method, we get, $$\begin{aligned} & \Rightarrow 6{{x}^{2}}-15x-8x+20=0 \\\ & \Rightarrow 3x\left( 2x-5 \right)-4\left( 2x-5 \right)=0 \\\ & \Rightarrow \left( 3x-4 \right)\left( 2x-5 \right)=0 \\\ \end{aligned}$$ Applying the zero-product property, we get, $$\Rightarrow $$ either $$x=\dfrac{4}{3}$$ or $$x=\dfrac{5}{2}$$ Therefore, $$x=\dfrac{4}{3}$$ or $$x=\dfrac{5}{2}$$ is the solution. (c) $$0={{x}^{2}}-9$$ $$\begin{aligned} & \Rightarrow {{x}^{2}}-9=0 \\\ & \Rightarrow {{x}^{2}}-{{3}^{2}}=0 \\\ \end{aligned}$$ Using the algebraic identity: - $${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$$, we get, $$\Rightarrow $$ either x = -3 or x = 3 Therefore, x = -3 or x = 3 is the solution. (d) $$0={{x}^{2}}+12x+36$$ $$\Rightarrow {{x}^{2}}+12x+36=0$$ This can be written as: - $$\Rightarrow {{x}^{2}}+2\times 6\times x+{{6}^{2}}=0$$ Using the algebraic identity: - $${{a}^{2}}+2ab+{{b}^{2}}={{\left( a+b \right)}^{2}}$$, we get, $$\begin{aligned} & \Rightarrow {{\left( x+6 \right)}^{2}}=0 \\\ & \Rightarrow \left( x+6 \right)\left( x+6 \right)=0 \\\ \end{aligned}$$ Applying the zero-product property, we get, $$\begin{aligned} & \Rightarrow x+6=0 \\\ & \Rightarrow x=-6 \\\ \end{aligned}$$ **Therefore, $x = -6$ is the solution.** **Note:** One must not apply the discriminant formula to solve the above questions because if you will do so then according to the question it will be considered as a wrong approach. This is because we have to factorize the equations and then use the zero-product property but the discriminant formula will directly give the values of x without factoring. You must remember the zero-product property, middle term split method and all the algebraic identities.