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Factoring Quadratics

Factoring Quadratics 1.10 - Solution

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We want to solve the given equation by factoring. To do so, we need to rewrite the linear coefficient -21\text{-} 21 as the sum of two numbers whose product is ac.{\color{#0000FF}{a}}{\color{#009600}{c}}. 2x221x36=0ac=2(-36)=-72\begin{gathered} {\color{#0000FF}{2}}x^2-21x{\color{#009600}{\,-\,36}}=0 \\ \Downarrow \\ {\color{#0000FF}{a}}{\color{#009600}{c}}={\color{#0000FF}{2}}({\color{#009600}{\text{-} 36}})=\text{-} 72 \end{gathered} Therefore, we need to find two numbers whose sum is -21\text{-} 21 and whose product is -72.\text{-} 72. In this case these numbers are -24\text{-} 24 and 3.3. With this, we can factor the left-hand side and solve the equation by using the Zero Product Property.

Factoring

Let's start by rewriting -21x\text{-} 21x as -24x+3x.\text{-} 24x+3x.
2x221x36=02x^2-21x-36=0
2x224x+3x36=02x^2-24x+3x-36=0
Factor out 2x & 32x\ \&\ 3
2x(x12)+3x36=02x(x-12)+3x-36=0
2x(x12)+3(x12)=02x(x-12)+3(x-12)=0
(x12)(2x+3)=0(x-12)(2x+3)=0

Solving

To solve this equation, we will apply the Zero Product Property.
(x12)(2x+3)=0(x-12)(2x+3)=0
x12=0(I)2x+3=0(II)\begin{array}{lc}x-12=0 & \text{(I)}\\ 2x+3=0 & \text{(II)}\end{array}
x=122x+3=0\begin{array}{l}x=12 \\ 2x+3=0 \end{array}
x=122x=-3\begin{array}{l}x=12 \\ 2x=\text{-}3 \end{array}
x1=12x2=-32\begin{array}{l}x_1=12 \\ x_2=\frac{\text{-} 3}{2} \end{array}
x1=12x2=-32\begin{array}{l}x_1=12 \\ x_2=\text{-} \frac{3}{2} \end{array}