# 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}=x^{2}+ 2xy+y^{2}(Square of a sum)(

x−y)^{2}=x^{2}− 2xy+y^{2}(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 *x*^{2}
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 *x*^{2} + 10*x* − 4 = 0 using completing the square method.

### Example 2

Solve 4*x*^{2} + *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`

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