Example 6: Solve the following equations by factoring. 1. x^2-4=0 2. 2x^2-18=0

To solve the equations by factoring, we need to factor the quadratic expressions and then set each factor equal to zero to find the solutions.<br /><br />1. $x^{2}-4=0$<br /><br />We can factor the left side of the equation as a difference of squares:<br />$x^{2}-4 = (x-2)(x+2) = 0$<br /><br />Setting each factor equal to zero gives us the solutions:<br />$x-2=0$ or $x+2=0$<br />$x=2$ or $x=-2$<br /><br />So, the solutions to the equation $x^{2}-4=0$ are $x=2$ and $x=-2$.<br /><br />2. $2x^{2}-18=0$<br /><br />We can factor out a common factor of 2 from the left side of the equation:<br />$2x^{2}-18 = 2(x^{2}-9) = 0$<br /><br />Now, we can factor the expression inside the parentheses as a difference of squares:<br />$2(x^{2}-9) = 2(x-3)(x+3) = 0$<br /><br />Setting each factor equal to zero gives us the solutions:<br />$x-3=0$ or $x+3=0$<br />$x=3$ or $x=-3$<br /><br />So, the solutions to the equation $2x^{2}-18=0$ are $x=3$ and $x=-3$.