Chapter 4 · Question 3

Explain the completing-the-square idea for solving a quadratic equation.

Answer

For a quadratic equation

ax^2+bx+c=0

with

a\ne0

, first divide by

a

and add and subtract

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

to form a perfect square. This gives

\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}

, after which taking square roots gives the solutions.

Divide by

a

, add and subtract

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

to make a square, then take square roots.

For equations of the type

x^2+2px+q=0

, add and subtract

p^2

:

x^2+2px+p^2-p^2+q=0

, so

(x+p)^2=p^2-q

. If the right side is non-negative, take square roots and solve for

x

. This method leads naturally to the quadratic formula.

Completing the square rewrites a quadratic expression as a perfect square plus or minus a constant, so the equation can be solved by taking square roots.