Roots by completing the square
Objective. Solve non-factorisable quadratics by completing the square.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Roots by completing the square solves a quadratic equation ax²+bx+c=0 by rewriting it in the form (x+p)²=q, then taking the square root of both sides to find exact or surd solutions, which works even when the quadratic does not factorise. For x²+bx+c, halve the coefficient of x to find p, then adjust the constant to keep the equation balanced, giving (x+b/2)² = (b/2)²−c. Taking square roots gives x = −p ± √q, which must include both the positive and negative root. For example, x²+6x+2=0 becomes (x+3)²=7, so x = −3±√7.
Worked example
Solve x²−4x−3=0 by completing the square, giving your answers in surd form.
- 1Halve the x-coefficient: half of −4 is −2, so write (x−2)²
- 2Expand (x−2)² = x²−4x+4, so x²−4x−3 = (x−2)²−4−3 = (x−2)²−7
- 3Set (x−2)²−7=0, so (x−2)²=7
- 4Take square roots: x−2 = ±√7, so x = 2±√7
Answer: x = 2+√7 or x = 2−√7
Concept mastery
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Practice
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Start practisingCommon mistake. Students often forget the ± when taking the square root of both sides, giving only one solution such as x = −3+√7 instead of both x = −3+√7 and x = −3−√7; a completed square always yields two roots unless q=0.
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