2. Solving Quadratic Equations by Completing the Square

For quadratic equations that cannot be solved by factorising, we use a method which can solve ALL quadratic equations called completing the square. We use this later when studying circles in plane analytic geometry.

Completing the square comes from considering the special formulas:

Follow these steps for the quadratic equation

ax² + bx + c = 0

(i) If a does not equal 1, divide each side by a (so that the coefficient of the x² is 1).

(ii) Rewrite the equation with the constant term on the right side.

(iii) Complete the square by adding the square of one-half of the coefficient of x to both sides.

(iv) Write the left side as a square and simplify the right side.

(v) Equate and solve.

Example:

Find the roots of x² + 10x − 4 = 0 using completing the square method.

Let's see the LiveMath solution first.

LIVEMath

Answer

Step (i) a = 1 [no action necessary]

Step (ii) Rewrite the equation with the constant term on the right side.

x² + 10x = 4

Step (iii) Complete the square by adding the square of one-half of the coefficient of x to both sides. In this case:

.

x² + 10x + 25 = 4 + 25

x² + 10x + 25 = 29

Step (iv) Write the left side as a square:

(x + 5)² = 29

Step (v) Equate and solve

Example

Solve 4x² + x = 3 by completing the square.

Answer

Exercises

Solve the following quadratic equations by completing the square

Q1. 2s² + 5s = 3

Q2. 3x²= 3 − 4x

Q3. 9v² − 6v − 2 = 0

Answer

1. Solving Quadratic Equations by Factoring

3. The Quadratic Formula

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