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The quadratic formula helps you solve quadratic equations, and is probably one of the top five formulas in math. We’re not big fans of you memorizing formulas, but this one is useful (and we think you should learn how to derive it as well as use it, but that’s for the second video!).
If you have a general quadratic equation like this:
ax2+bx+c=0ax^2+bx+c=0ax2+bx+c=0a, x, squared, plus, b, x, plus, c, equals, 0
Then the formula will help you find the roots of a quadratic equation, i.e. the values of x x xx where this equation is solved.
The quadratic formula
x=−b±b2−4ac2ax=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}x=2a−b±b2−4ac
Worked example
First we need to identify the values for a, b, and c (the coefficients). First step, make sure the equation is in the format from above, ax2+bx+c=0 ax^2 + bx + c = 0 ax2+bx+c=0a, x, squared, plus, b, x, plus, c, equals, 0:
x2+4x−21=0x^2+4x-21=0x2+4x−21=0x, squared, plus, 4, x, minus, 21, equals, 0
a a aa is the coefficient in front of x2 x^2 x2x, squared, so here a=1 a = 1 a=1a, equals, 1 (note that a a aa can’t equal 0 0 00 -- the x2 x^2 x2x, squared is what makes it a quadratic).
b b bb is the coefficient in front of the x x xx, so here b=4 b = 4 b=4b, equals, 4.
c c cc is the constant, or the term without any x x xx next to it, so here c=−21 c = -21 c=−21c, equals, minus, 21.
Then we plug a a aa, b b bb, and c c cc into the formula:
x=−4±16−4⋅1⋅(−21)2x=\dfrac{-4\pm\sqrt{16-4\cdot 1\cdot (-21)}}{2}x=2−4±16−4⋅1⋅(−21)
x, equals, start fraction, minus, 4, plus minus, square root of, 16, minus, 4, dot, 1, dot, left parenthesis, minus, 21, right parenthesis, end square root, divided by, 2, end fraction
solving this looks like:
x=−4±1002=−4±102=−2±5\begin{aligned} x&=\dfrac{-4\pm\sqrt{100}}{2} \\\\ &=\dfrac{-4\pm 10}{2} \\\\ &=-2\pm 5 \end{aligned}x=2−4±100
=2−4±10=−2±5
Therefore x=3 x = 3 x=3x, equals, 3 or x=−7 x = -7 x=−7x, equals, minus, 7.
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