examples: solve 12x^2 - 2x = x by factoring. solve x^2 - 6x - 9 = 18 by factoring. solve 3x^2 + 5x + 15 = 17…

Explanation:

Step1: Rearrange the equation to standard quadratic form

For the equation $12x^{2}-2x = x$, we move all terms to one - side: $12x^{2}-2x - x=0$, which simplifies to $12x^{2}-3x = 0$. Then factor out the greatest common factor. Here, the GCF of $12x^{2}$ and $-3x$ is $3x$, so we have $3x(4x - 1)=0$. Using the zero - product property $ab = 0$ implies $a = 0$ or $b = 0$, we get $3x=0$ or $4x - 1=0$. Solving $3x=0$ gives $x = 0$, and solving $4x - 1=0$ gives $x=\frac{1}{4}$. For the equation $x^{2}-6x - 9 = 18$, we move 18 to the left - hand side to get $x^{2}-6x-9 - 18=0$, which simplifies to $x^{2}-6x - 27=0$. We need to find two numbers that multiply to $-27$ and add up to $-6$. The numbers are $-9$ and $3$. So we can factor the quadratic as $(x - 9)(x+3)=0$. Using the zero - product property, we have $x - 9=0$ or $x + 3=0$, so $x = 9$ or $x=-3$. For the equation $3x^{2}+5x + 15 = 17$, we move 17 to the left - hand side: $3x^{2}+5x+15 - 17=0$, which simplifies to $3x^{2}+5x - 2=0$. We need to find two numbers that multiply to $3\times(-2)=-6$ and add up to $5$. The numbers are $6$ and $-1$. Rewrite the middle term: $3x^{2}+6x - x - 2=0$. Group the terms: $(3x^{2}+6x)-(x + 2)=0$. Factor out the GCF from each group: $3x(x + 2)-(x + 2)=0$. Then $(3x - 1)(x + 2)=0$. Using the zero - product property, we get $3x - 1=0$ or $x + 2=0$, so $x=\frac{1}{3}$ or $x=-2$.

Answer:

For $12x^{2}-2x = x$, $x = 0$ or $x=\frac{1}{4}$; for $x^{2}-6x - 9 = 18$, $x = 9$ or $x=-3$; for $3x^{2}+5x + 15 = 17$, $x=\frac{1}{3}$ or $x=-2$