Quadratic formula review

The quadratic formula allows us to solve any quadratic equation that's in the form ax^2 + bx + c = 0. This article reviews how to apply the formula.

What is the quadratic formula?

The quadratic formula says that
x=b±b24ac2ax=\dfrac{-\goldD{b}\pm\sqrt{\goldD{b}^2-4\purpleD{a}\redD{c}}}{2\purpleD{a}}
for any quadratic equation like:
ax2+bx+c=0\purpleD{a}x^2 + \goldD{b}x + \redD{c} = 0

Example

We're given an equation and asked to solve for qq:
0=7q2+2q+90=-7q^2+2q+9
This equation is already in the form ax2+bx+c=0ax^2 + bx + c = 0, so we can apply the quadratic formula where a=7,b=2,c=9a = -7, b = 2, c = 9:
q=b±b24ac2aq=2±224(7)(9)2(7)q=2±4+25214q=2±25614q=2±1614q=2+1614  ,  q=21614q=1            ,  q=97\begin{aligned} q &= \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\\\ q &= \dfrac{-2 \pm \sqrt{2^{2} - 4 (-7) (9)}}{2(-7)} \\\\ q &= \dfrac{-2 \pm \sqrt{4 +252}}{-14} \\\\ q &= \dfrac{-2 \pm \sqrt{256}}{-14} \\\\ q &= \dfrac{-2 \pm 16}{-14} \\\\ q &= \dfrac{-2 + 16}{-14} ~~,~~ q = \dfrac{-2 - 16}{-14} \\\\ q &= -1 ~~~~~~~~~~~~,~~ q = \dfrac{9}{7} \end{aligned}
Let's check both solutions to be sure it worked:
q=1q=-1q=97q=\dfrac{9}{7}
0=7q2+2q+90=7(1)2+2(1)+90=7(1)2+90=72+90=0\begin{aligned}0&=-7q^2+2q+9\\\\0&=-7(-1)^2+2(-1)+9 \\\\0&=-7(1)-2+9 \\\\0&=-7-2+9\\\\0&=0\end{aligned}0=7q2+2q+90=7(97)2+2(97)+90=7(8149)+(187)+90=(817)+(187)+90=(637)+90=9+90=0\begin{aligned}0&=-7q^2+2q+9\\\\0&=-7\left(\dfrac{9}{7}\right)^2+2\left (\dfrac{9}{7}\right)+9 \\\\0&=-7\left(\dfrac{81}{49}\right)+\left (\dfrac{18}{7}\right)+9 \\\\0&=-\left(\dfrac{81}{7}\right)+\left (\dfrac{18}{7}\right)+9 \\\\0&=-\left(\dfrac{63}{7}\right) +9 \\\\0&=-9 +9 \\\\0&=0\end{aligned}
Yep, both solutions check out.
Want to learn more about the quadratic formula? Check out this video.
Want more practice? Check out this exercise.
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