How to Solve a Quadratic Equation by Completing the Square?

When we want to convert a quadratic expression of the form \(ax^2+ bx+c\) to the vertex form, we use the completing square method.

Related Topics

• How to Solve a Quadratic Equation by Factoring

A step-by-step guide to completing the square

Completing a square is a method used to convert a quadratic expression of the form \(ax^2+ bx+c\) to the vertex form \(a(x-h)^2+k\). The most common application of completing the square is in solving a quadratic equation. This can be done by rearranging the expression obtained after completing the square: \(a(x + m)^2+n\), so that the left side is a perfect square trinomial.

Completing the square method is useful in the following cases:

• Converting a quadratic expression into vertex form.
• Analyzing at which point the quadratic expression has the minimum or maximum value.
• Graphing a quadratic function.
• Solving a quadratic equation.
• Deriving the quadratic formula.

Completing the square method

The most common application of completing the square method is factorizing a quadratic equation, and henceforth finding the roots and zeros of a quadratic polynomial or a quadratic equation. We know that a quadratic equation in the form of \(ax^2+bx+c=0\) can be solved by the factorization method. But sometimes, factorization of the quadratic expression \(ax^2+bx+c\) is complex or impossible.

Completing the square formula

Completing a square formula is a method for converting a quadratic polynomial or equation to a complete square with an additional constant value. A quadratic expression in variable \(x\): \(ax^2+ bx+ c\), where \(a, b\) and \(c\) are any real numbers but \(a≠0\), can be converted into a perfect square with some additional constant by using completing the square formula or technique.

Completing the square formula is a method that can be used to find the roots of the given quadratic equations, \(ax^2+bx+c\), where \(a, b\) and \(c\) are any real numbers but \(a≠0\).

The formula for completing the square:

The formula for completing the square is:

\(\color{blue}{ax^2 + bx + c ⇒ a(x + m)^2+ n}\)

where:

• \(m\) is any real number
• \(n\) is a constant term

Instead of using the complicated step-by-step method to complete the square, we can use the simple formula below to complete the square. To complete the square in the expression \(ax^2+bx + c\), first find:

\(\color{blue}{m= \frac{b}{2a}}\) , \(\color{blue}{n=c – (\frac{b^2}{4a})}\)

Substitute these values in: \(ax^2+bx +c = a(x + m)^2+n\). These formulas are derived geometrically.

Exercises for Completing the Square

Solve each equation by completing the square. 

1. \(\color{blue}{x^2+12x+32=0}\)
2. \(\color{blue}{x^2-6x-3=0}\)
3. \(\color{blue}{x^2-10x+16=0}\)
4. \(\color{blue}{2x^2+7x+6=0}\)

1. \(\color{blue}{x=-4, -8}\)
2. \(\color{blue}{x=3+2\sqrt{3},\:3-2\sqrt{3}}\)
3. \(\color{blue}{x=2,8}\)
4. \(\color{blue}{x=-\frac{3}{2}, -2}\)