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 that we met in Square of a sum and square of a difference earlier:

(x + y)2 = x2 + 2xy + y2 (Square of a sum)

(xy)2 = x2 − 2xy + y2 (Square of a difference)

To find the roots of a quadratic equation in the form:

`ax^2+ bx + c = 0`,

follow these steps:

(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 1

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

Example 2

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

Exercises

Solve the following quadratic equations by completing the square

Q1. `2s^2+ 5s = 3`

Q2. `3x^2= 3 − 4x`

Q3. `9v^2− 6v − 2 = 0`

Q4. `ax^2+bx+c=0`