Example 7: Solve the following equations by factoring. 1 9x^2-6x+1=0 2. x^2+x-12=0 9x^2-6x+1=(3x-1)^2

Let's solve the given equations by factoring.<br /><br />1. $9x^{2}-6x+1=0$<br /><br />To solve this equation by factoring, we need to find two numbers whose product is equal to the product of the coefficient of $x^2$ (which is 9) and the constant term (which is 1), and whose sum is equal to the coefficient of $x$ (which is -6).<br /><br />The two numbers that satisfy these conditions are -3 and -1. Therefore, we can rewrite the equation as:<br /><br />$(3x-1)^{2}=0$<br /><br />Now, we can set each factor equal to zero and solve for $x$:<br /><br />$3x-1=0$<br /><br />Solving for $x$, we get:<br /><br />$x=\frac{1}{3}$<br /><br />So, the solution to the equation $9x^{2}-6x+1=0$ is $x=\frac{1}{3}$.<br /><br />2. $x^{2}+x-12=0$<br /><br />To solve this equation by factoring, we need to find two numbers whose product is equal to the product of the coefficient of $x^2$ (which is 1) and the constant term (which is -12), and whose sum is equal to the coefficient of $x$ (which is 1).<br /><br />The two numbers that satisfy these conditions are 4 and -3. Therefore, we can rewrite the equation as:<br /><br />$(x+4)(x-3)=0$<br /><br />Now, we can set each factor equal to zero and solve for $x$:<br /><br />$x+4=0$ or $x-3=0$<br /><br />Solving for $x$, we get:<br /><br />$x=-4$ or $x=3$<br /><br />So, the solutions to the equation $x^{2}+x-12=0$ are $x=-4$ and $x=3$.