Quadratic Equation by Quadratic Formula
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*}
Example 4
$$\textup{Solve } x^2+10x+7=0$$
Solution
\begin{align*}
& x^2+10x+7= 0\\\\
\because \;\;\; & ax^2+bx+c=0\\
\therefore \;\;\; & a=1, b=10, c=7\\\\
\because x &= \frac{-b \pm \sqrt{b^2-4ac}}{2a}\\
\therefore x &= \frac{-10 \pm \sqrt{(10)^2-4(1)(7)}}{2(1)}\\
x &= \frac{-10 \pm \sqrt{100-28}}{2}\\
x &= \frac{-10 \pm \sqrt{72}}{2}\\
x &= \frac{-10 \pm \sqrt{(36)(2)}}{2}\\
x &= \frac{-10 \pm \sqrt{36} \sqrt{2}}{2}\\
x &= \frac{-10 \pm 6 \sqrt{2}}{2}\\
x &= \frac{\cancel{2} (-5 \pm 3 \sqrt{2})}{\cancel{2}}\\
x &= -5 \pm 3 \sqrt{2}\\
x &= -5 + 3 \sqrt{2}, \;\;\; x = -5 - 3\sqrt{2}\\\\
\end{align*}