We will explain how completing the square is used to derive the Quadratic Formula. We can consider a quadratic equation written in standard form, complete the square, and derive the formula by ourselves. Let's start by writing the constant term

c

on the right-hand side and dividing the equation by the leading coefficient

a .

a x^{2} + b x + c = 0

SubEqn

$LHS - c = RHS - c$

a x^{2} + b x = - c

DivEqn

$LHS / a = RHS / a$

\frac{a x ^{2} + b x}{a} = \frac{- c}{a}

▼

Simplify left-hand side

WriteSumFrac

Write as a sum of fractions

\frac{a x ^{2}}{a} + \frac{b x}{a} = \frac{- c}{a}

MovePartNumRight

$\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

\frac{a}{a} x^{2} + \frac{b}{a} x = \frac{- c}{a}

QuotOne

$\frac{a}{a} = 1$

1 x^{2} + \frac{b}{a} x = \frac{- c}{a}

OneMult

$1 \cdot a = a$

x^{2} + \frac{b}{a} x = \frac{- c}{a}

Next, we want the left-hand side of the above equation to be a perfect square trinomial. For this to happen, we need three terms. The first and third terms must be perfect squares and the second term must be twice the product of the square root of the first and third terms. To achieve this, we will add

(\frac{b}{2 a})^{2}

to both sides.

x^{2} + \frac{b}{a} x = \frac{- c}{a}

AddEqn

$LHS + (\frac{b}{2 a})^{2} = RHS + (\frac{b}{2 a})^{2}$

x^{2} + \frac{b}{a} x + (\frac{b}{2 a})^{2} = \frac{- c}{a} + (\frac{b}{2 a})^{2}

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