What if the term $(b/2)^2$ results in a fraction?

Leave it as an exact fraction. Do not convert it to a decimal, as fractions make factoring the perfect square and taking the square root significantly easier and prevent rounding errors.

What if the right side of the equation is negative after adding $(b/2)^2$?

The equation has complex roots. Taking the square root of the negative number on the right side will introduce the imaginary unit $i$.

Can I use completing the square if $b=0$?

Yes, but it is inefficient. If $b=0$, the equation is already set up to be solved by directly isolating $x^2$ and taking the square root. The $(b/2)^2$ term evaluates to zero.

How to solve a quadratic by completing the square

Completing the square transforms a standard quadratic equation $ax^2 + bx + c = 0$ into the form $a(x-h)^2 + k = 0$ , allowing direct solution by taking square roots. This method applies to all quadratic equations, including those with complex, irrational, or rational roots.

The setup

Ensure the equation is in the standard form $ax^2 + bx + c = 0$ . Divide the entire equation by the leading coefficient $a$ so that the coefficient of the $x^2$ term is exactly $1$ . Move the constant term to the right side of the equality.

The steps

Identify the new linear coefficient $b$ in the equation $x^2 + bx = c$ .
Compute the value of $(b/2)^2$ .
Add this computed value to both sides of the equation.
Factor the left side as the perfect square $(x + b/2)^2$ .
Take the square root of both sides, applying the $\pm$ sign to the right side.
Isolate $x$ to find the roots.

Checking the result

Substitute both solutions back into the original unsimplified equation $ax^2 + bx + c = 0$ . Both values must satisfy the equality. If they do not, check for arithmetic errors in factoring or evaluating the square root.

Common errors

Forgetting to divide by $a$ before calculating $(b/2)^2$ . Neglecting the $\pm$ symbol when taking the square root, which incorrectly drops one of the two solutions. Adding the squared term to only the left side of the equation, violating algebraic equality.

Worked example

Solve $2x^2 - 8x - 10 = 0$ by completing the square.

$2x^2 - 8x - 10 = 0$ Divide by the leading coefficient 2: $x^2 - 4x - 5 = 0$ Move the constant to the right side: $x^2 - 4x = 5$ Find $(b/2)^2$ where $b = -4$ : $(-4/2)^2 = (-2)^2 = 4$ Add 4 to both sides: $x^2 - 4x + 4 = 5 + 4$ $x^2 - 4x + 4 = 9$ Factor the left side as a perfect square: $(x - 2)^2 = 9$ Take the square root of both sides: $x - 2 = \pm 3$ Solve for $x$ : $x = 2 \pm 3$ Calculate the two roots: $x_1 = 2 + 3 = 5$ $x_2 = 2 - 3 = -1$

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References: OpenStax College Algebra, Section 2.5: Quadratic Equations · Khan Academy Unit: Quadratic Equations & Functions · Stewart Precalculus, Chapter 1: Fundamentals