"Where does the plus-minus come from in the..." was answered by our community

High School
Algebra
Algebra I
Quadratic function and equation

Samir Duffy

Answered question

2022-03-15

Where does the plus-minus come from in the quadratic formula?

x_{1, 2} = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

Answer & Explanation

Corbin Pittman

Beginner2022-03-16Added 4 answers

Step 1
$a x^{2} + b x + c = a (x^{2} + \frac{b x}{a}) + c$
$= a (x^{2} + \frac{b x}{a} + \frac{b^{2}}{4 a^{2}}) + c - \frac{b^{2}}{4 a}$
$= a {(x + \frac{b}{2 a})}^{2} + \frac{4 a c - b^{2}}{4 a}$
Hence we deduce that
$a x^{2} + b x + c = 0 ⟺ a (x + \frac{b}{2 a})^{2} + \frac{4 a c - b^{2}}{4 a} = 0$
$⟺ (x + \frac{b}{2 a})^{2} = \frac{b^{2} - 4 a c}{4 a^{2}}$
$⟺ x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
and we are done.

Loyau97

Beginner2022-03-17Added 2 answers

Step 1
Given a general quadratic
$a x^{2} + b x + c = 0$
Multiply through by 4a, add $b^{2} &$ this will allow us to "complete the square"
$4 a^{2} x^{2} + 4 a b x + b^{2} = b^{2} - 4 a c$
${(2 a x + b)}^{2} = b^{2} - 4 a c$
Now take the square root, this is where the plus-minus comes from.

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