Quadratic Equation by Quadratic Formula

Example 1

$$\textup{Solve } x^2+7x+4=0$$

Solution

\begin{align*} & x^2+7x+4 = 0\\\\ \because \;\;\; & ax^2+bx+c=0\\ \therefore \;\;\; & a=1, b=7, c=4\\\\ \because x &= \frac{-b \pm \sqrt{b^2-4ac}}{2a}\\ \therefore x &= \frac{-7 \pm \sqrt{(7)^2-4(1)(4)}}{2(1)}\\ x &= \frac{-7 \pm \sqrt{49-16}}{2}\\ x &= \frac{-7 \pm \sqrt{33}}{2}\\ x &= \frac{-7 + \sqrt{33}}{2}, \;\;\; x = \frac{-7 - \sqrt{33}}{2}\\\\ \end{align*}

Example 2

$$\textup{Solve } x^2-5x+3=0$$

Solution

\begin{align*} & x^2-5x+3= 0\\\\ \because \;\;\; & ax^2+bx+c=0\\ \therefore \;\;\; & a=1, b=-5, c=3\\\\ \because x &= \frac{-b \pm \sqrt{b^2-4ac}}{2a}\\ \therefore x &= \frac{-(-5) \pm \sqrt{(-5)^2-4(1)(3)}}{2(1)}\\ x &= \frac{5 \pm \sqrt{25-12}}{2}\\ x &= \frac{5 \pm \sqrt{13}}{2}\\ x &= \frac{5 + \sqrt{13}}{2}, \;\;\; x = \frac{5 - \sqrt{13}}{2}\\\\ \end{align*}

Example 3

$$\textup{Solve } x^2+6x+4=0$$

Solution

\begin{align*} & x^2+6x+4= 0\\\\ \because \;\;\; & ax^2+bx+c=0\\ \therefore \;\;\; & a=1, b=6, c=4\\\\ \because x &= \frac{-b \pm \sqrt{b^2-4ac}}{2a}\\ \therefore x &= \frac{-6 \pm \sqrt{(6)^2-4(1)(4)}}{2(1)}\\ x &= \frac{-6 \pm \sqrt{36-16}}{2}\\ x &= \frac{-6 \pm \sqrt{20}}{2}\\ x &= \frac{-6 \pm \sqrt{(4)(5)}}{2}\\ x &= \frac{-6 \pm \sqrt{4} \sqrt{5}}{2}\\ x &= \frac{-6 \pm 2 \sqrt{5}}{2}\\ x &= \frac{\cancel{2} (-3 \pm \sqrt{5})}{\cancel{2}}\\ x &= -3 \pm \sqrt{5}\\ x &= -3 + \sqrt{5}, \;\;\; x = -3 - \sqrt{5}\\\\ \end{align*}