Example 1: Solve $3x^2+x−14=0$ for $x$ using the Zero Product Property.

Solution: First, factor the quadratic so it is written as a product: $(3x+7)(x−2)=0$. (If factoring is not possible, one of the other methods of solving must be used.)  The Zero Product Property states that if the product of two terms is $0$, then at least one of the factors must be $0$.  Thus, $3x+7=0$ or $x−2=0$. Solving these equations for $x$ reveals that $x=\frac{7}{3}$ or that $x=2$.

Example 2: Solve $3x^2+x−14=0$ for $x$ using the Quadratic Formula.

Solution: This method works for any quadratic.  First, identify $a,b$, and $c$.  $a$ equals the number of $x^2$-terms, $b$ equals the number of $x$-terms, and $c$ equals the constant. For $3x^2+x−14=0,a=3,b=1$, and $c=−14$.  Substitute the values of $a,b$, and $c$ into the Quadratic Formula and evaluate the expression twice: once with addition and once with subtraction. Examine this method below:

$\begin{array}{l} x = \frac{-1 + \sqrt{1^2 - 4\left(3\right) \left(-14\right)}}{2 \cdot 3} \\ \phantom{x} = \frac{-1 + \sqrt{169}}{6} \\ \phantom{x} = \frac{12}{6} \\ \phantom{x} = 2 \end{array} \quad \text{or} \quad \begin{array}{l} x = \frac{-1 + \sqrt{1^2 - 4\left(3\right) \left(-14\right)}}{2 \cdot 3} \\ \phantom{x} = \frac{-1 - \sqrt{169}}{6} \\ \phantom{x} = \frac{-14}{6} \\ \phantom{x} = -\frac{7}{3} \end{array}$

Example 3: Solve $x^2+5x+4=0$ by completing the square.

Solution: This method works most efficiently when the coefficient of $x^2$ is $1$.  Rewrite the equation as $x^2+5x=-4$.  Rewrite the left side as an incomplete square:

Take the square root of both sides, $x+2.5=\pm1.5$.  Solving for $x$ reveals that $x=-1$ or $x=-4$.