Solving Quadratic Equations: Completing the Square

The core formula for completing the square is:

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

• If the coefficient is negative, the bracket will be $(x - p)^2$. Note that the square of a negative number is positive, but you must still subtract it outside the bracket.
• If the coefficient is odd, leave the halved value as a fraction (e.g., $\frac{3}{2}$) rather than a decimal, as it makes squaring much easier (e.g., ).

Solving Quadratic Equations ($a = 1$)

Before you can solve, the equation must be set to zero. This may require quadratic rearrangement to get all terms onto one side in the form $x^2 + bx + c = 0$. Once the square is completed, isolate the bracket and take the square root of both sides.

• When square-rooting, you must include the $\pm$ symbol to account for both the positive and negative root.
• Edexcel Higher tier papers frequently specify that answers must be left in exact form or surd form (e.g., $x = a \pm b\sqrt{c}$) rather than decimals.

Worked Example 1: Solving and Surd Simplification

Solve $2x^2 - 11x - 25 = x^2 - x$ by completing the square. Give your answer in simplified surd form.

Step 1: Rearrange the equation into the standard form $ax^2 + bx + c = 0$.

• $x^2 - 10x - 25 = 0$

Step 2: Halve the $x$-coefficient ($-10 \div 2 = -5$) to create the perfect square, then subtract its square.

• $(x - 5)^2 - (-5)^2 - 25 = 0$

Step 3: Simplify the constant terms.

• $(x - 5)^2 - 25 - 25 = 0$
• $(x - 5)^2 - 50 = 0$

Step 4: Rearrange to isolate the squared bracket.

• $(x - 5)^2 = 50$

Step 5: Square root both sides, remembering the $\pm$ symbol.

• $x - 5 = \pm\sqrt{50}$

Step 6: Simplify the surd by extracting the largest square factor of $50$ ($\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$).

• $x - 5 = \pm 5\sqrt{2}$

Step 7: Solve for $x$.

• $x = 5 \pm 5\sqrt{2}$

Handling Non-Unitary Coefficients

A non-unitary coefficient occurs when the number in front of the $x^2$ term is not $1$ (for example, $2x^2$ or $-x^2$). If you are solving an equation like , the easiest approach is to divide every single term by first to simplify the process back to the method.

Worked Example 2: Solving with $a > 1$

Solve $2x^2 + 12x + 4 = 0$ by completing the square. Leave your answer in exact form.

Step 1: Divide the entire equation by the non-unitary coefficient ($2$).

• $x^2 + 6x + 2 = 0$

Step 2: Halve the $x$-coefficient ($6$) and complete the square.

• $(x + 3)^2 - 3^2 + 2 = 0$

Step 3: Simplify the constants.

• $(x + 3)^2 - 9 + 2 = 0$
• $(x + 3)^2 - 7 = 0$

Step 4: Isolate the bracket and square root both sides.

• $(x + 3)^2 = 7$
• $x + 3 = \pm\sqrt{7}$

Step 5: Solve for $x$.

• $x = -3 \pm \sqrt{7}$