Solving Quadratic Equations by Completing the Square
July 26, 2025
Introduction
A quadratic equation is a polynomial equation of the form:
where a, \, b, \, \text{ and } c are real number and a \neq 0.
Completing the square transforms a quadratic
into a perfect-square trinomial, allowing us to solve by taking square roots:
To ensure you apply it correctly every time, follow these simple guidelines.
Follow these steps to use completing the square in solve:
- Ensure that the quadratic equation is in standard form:
- Transpose the constant c to the right-hand side, thus obtaining:
- Add both sides of the equation by the quantity \dfrac{b^2}{4a} to make sure that the left hand side is a perfect square trinomial:
- Factor the left hand side to a square of a binomial. Form that point, proceed to extracting the roots.
- Simplify the solved values of x, if necessary.
Illustrative Example #1
Find the solutions of the equation x^2-2x-3=0 using completing the square.
Step 1: The equation is already in standard form:
Step 2: Transpose the constant to the right side:
Step 3: Compute \displaystyle \dfrac{b^2}{4a}=\dfrac{(-2)^2}{4\cdot1} =1 and add to both sides:
Step 4: Factor the left-hand side and simplify the right:
Step 5: Take square roots of both sides:
Step 6: Solve for x:
Therefore, the solutions are
Illustrative Example #2
Find the solutions of the equation y^2+6y-59=0 using completing the square.
Step 1: The equation is already in standard form:
Step 2: Transpose the constant to the right side:
Step 3: Compute \displaystyle \frac{b^2}{4a} = \frac{6^2}{4\cdot1} = 9 and add to both sides:
Step 4: Factor the left-hand side and simplify the right:
Step 5: The square roots of both sides:
Step 6: Solve for y:
Therefore, the solutions are
Illustrative Example #3
Find the solutions of the equation m^2+14m-51=0 using completing the square.
Step 1: The equation is already in standard form:
Step 2: Transpose the constant to the right side:
Step 3: Compute \displaystyle \frac{b^2}{4a} = \frac{14^2}{4\cdot1} = 49 and add to both sides:
Step 4: Factor the left-hand side and simplify the right:
Step 5: Take square roots of both sides:
Step 6: Solve for m:
Therefore, the solutions are
Illustrative Example #4
Find the solutions of the equation u^2-10u+26=8 using completing the square.
Step 1: Transpose 8 to the left hand side to make it in standard form:
Step 2: Transpose the constant term 18 to the right side:
Step 3: Compute \displaystyle \frac{b^2}{4a} = \frac{(-10)^2}{4\cdot1} = 25 and add to both sides:
Step 4: Factor the left-hand side and take square roots of both sides:
Step 5: Solve for u:
Illustrative Example #5
Find the solutions of the equation \bm{5k^2 = 60 - 20k} using completing the square.
Step 1: Rewrite in standard form by equation it to 0:
Step 2: Transpose the constant to the right side:
Step 3: Compute \displaystyle \dfrac{b^2}{4a} = \dfrac{20^2}{4\cdot5} = 20 and add to both sides:
Step 4: Factor the left-hand side and simplify the right:
Step 5: Divide both sides by 5:
Step 6: Take square roots:
Step 7: Solve for \bm{k}:
Illustrative Example #6
Find the solutions of the equation \bm{6x^2 - 48 = -12x} using completing the square.
Step 1: Rewrite in standard form:
Step 2: Transpose the constant to the right side:
Step 3: Compute \displaystyle \dfrac{b^2}{4a} = \dfrac{12^2}{4\cdot6} = 6 and add to both sides:
Step 4: Factor the left-hand side and simplify the right:
Step 5: Divide both sides by 6:
Step 6: Take square roots:
Step 7: Solve for \bm{x}:
