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The Quadratic Formula

≈ 35 min
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Completing the square on the general quadratic ax2+bx+c=0ax^2 + bx + c = 0 once and for all produces the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

It solves every quadratic — factorisable or not.

Worked example 1. x2−5x+6=0x^2 - 5x + 6 = 0 (a=1,b=−5,c=6a=1, b=-5, c=6):

x=5±25−242=5±12  ⇒  x=3 or x=2x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2} \;\Rightarrow\; x = 3 \text{ or } x = 2

(The same roots factorising gives — reassuring.)

Worked example 2. 2x2+3x−2=02x^2 + 3x - 2 = 0 (a=2,b=3,c=−2a=2, b=3, c=-2):

x=−3±9+164=−3±54  ⇒  x=12 or x=−2x = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm 5}{4} \;\Rightarrow\; x = \tfrac{1}{2} \text{ or } x = -2

Use the quadratic formula to solve x2−5x+6=0x^2 - 5x + 6 = 0. Answer in the form x=… or x=…

Calculate the value under the square root (b2−4acb^2 - 4ac) for x2−4x+1=0x^2 - 4x + 1 = 0.

Solve with the formula: 2x2+3x−2=02x^2 + 3x - 2 = 0. Answer in the form x=… or x=… (fractions like 1/2 are fine).

Which is the quadratic formula?