Solving Quadratic Equations Using Factoring

To solve an quadratic equation using factoring :

$1$ . Transform the equation using standard form in which one side is zero.

$2$ .  Factor the non-zero side.

$3$ .  Set each factor to zero (Remember: a product of factors is zero if and only if one or more of the factors is zero).

$4$ .  Solve each resulting equation.

Example 1:

Solve the equation, ${x}^{2}-3x-10=0$

Factor the left side: $\left(x-5\right)\left(x+2\right)=0$

Set each factor to zero: $x-5=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x+2=0$

Solve each equation: $x=5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=-2$

The solution set is $\left\{5,-2\right\}$ .

Example 2:

Solve the equation, $2{x}^{2}+5x=12$

Set the right side to zero: $2{x}^{2}+5x-12=0$

Factor the left side: $\left(2x-3\right)\left(x+4\right)=0$

Set each factor to zero: $2x-3=0$ or $x+4=0$

Solve each equation: $x=\frac{3}{2}$ or $x=-4$

The solution set is $\left\{\frac{3}{2},-4\right\}$ .