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# 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, equals, start fraction, minus, start color #e07d10, b, end color #e07d10, plus minus, square root of, start color #e07d10, b, end color #e07d10, squared, minus, 4, start color #7854ab, a, end color #7854ab, start color #e84d39, c, end color #e84d39, end square root, divided by, 2, start color #7854ab, a, end color #7854ab, end fraction
for any quadratic equation like:
start color #7854ab, a, end color #7854ab, x, squared, plus, start color #e07d10, b, end color #e07d10, x, plus, start color #e84d39, c, end color #e84d39, equals, 0

### Example

We're given an equation and asked to solve for q:
0, equals, minus, 7, q, squared, plus, 2, q, plus, 9
This equation is already in the form a, x, squared, plus, b, x, plus, c, equals, 0, so we can apply the quadratic formula where a, equals, minus, 7, comma, b, equals, 2, comma, c, equals, 9:
\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, equals, minus, 1q, equals, start fraction, 9, divided by, 7, end fraction
\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}\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.
Practice
Solve for x.
minus, 4, plus, x, plus, 7, x, squared, equals, 0

Want more practice? Check out this exercise.

## Want to join the conversation?

• Sal, How does the quadratic formula relate to business and economics?
• It helps in lots of ways. It can possibly predict the future path of certain things, especially if your graph is exponential.
• what if the equation doesn't equal zero
• then just subtract the non-zero number from the RHS to the LHS and make the RHS equal to zero.
• Can u ask Sal to watch an education rap, it's called:

Bring It Back
by Balistik (ZT)

There's a cool little bar about education= the key to success
• can you reccomend other math websites for algebra 1 and 2
• What happens when the discriminant is a negative number? If it is negative would your answer be imaginary?
• Yes... If the discriminant is negative, then there are 2 roots, but they are complex numbers.
• If a quadratic equation ax^2+bx+c=0 has more than two roots, then it becomes an identity a=b=c=0.

Can someone please prove this above statement to me?
- How did the identity part come about? How can a=b=c=0?
(1 vote)
• First, in order for an equation to be a quadratic, the "a" (in ax^2+bx+c = 0) can't = 0
As soon as "a = 0", you no longer have a 2nd degree equation. And, your graph will not be a parabola.

If a=b=c=0, then the left side of the equation becomes 0. Do the substitution and you'll see.
Your equation becomes 0=0. This is an equation that is always true, and is called an identity. But, again, it is no longer a quadratic. "a" will never = 0 in a quadratic equation. Quadratics only have 0, 1, or 2 real roots. If they have 0 real roots, then they will have 2 complex roots.
• What happens when a discriminant is a negative number? If it is negative would your answer be imaginary?