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Factoring Quadratics

Factoring Quadratics 1.16 - Solution

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First we want to factor the quadratic trinomial 12x22x2.12x^2-2x-2. Let's start by factoring out a GCF, if we find one.
12x22x212x^2-2x-2
2(6x2x1)2(6x^2-x-1)
2(6x23x+2x1)2(6x^2-3x+2x-1)
2(3x(2x1)+2x1)2 \big( 3x(2x-1)+2x-1 \big)
2(3x+1)(2x1)2(3x+1)(2x-1)
We can now use this to write the given equation as 2(3x+1)(2x1)=0.2(3x+1)(2x-1)=0. To solve this equation, we will apply the zero product property.
2(3x+1)(2x1)=02(3x+1)(2x-1)=0
(3x+1)(2x1)=0(3x+1)(2x-1)=0
3x+1=0(I)2x1=0(II)\begin{array}{lc}3x+1=0 & \text{(I)}\\ 2x-1=0 & \text{(II)}\end{array}
3x=-12x1=0\begin{array}{l}3x=\text{-} 1 \\ 2x-1=0 \end{array}
x=-132x1=0\begin{array}{l}x=\text{-} \frac{1}{3} \\ 2x-1=0 \end{array}
x=-132x=1\begin{array}{l}x=\text{-} \frac{1}{3} \\ 2x=1 \end{array}
x1=-13x2=12\begin{array}{l}x_1=\text{-} \frac{1}{3} \\ x_2=\frac{1}{2} \end{array}