Question

express following expression in the form : (x+a)^+b (a): x^-6x+1

Answer 1

Answer:

Unit 17 Section 3 : Quadratic Equations: Completing the Square

Completing the square is a useful technique for solving quadratic equations. It is a more powerful technique than factorisation because it can be applied to equations that do not factorise.

When completing the square, an expression like,

ax2 + bx + c is written in the form (Ax + B)2 + C.

We will begin with the simple example where a = 1. In this case we will write expressions in the form

x2 + bx + c as (x + B)2 + C

If we expand (x + B)2 + C we get x2 + 2Bx + B2 + C.

Comparing this with x2 + bx + c shows that

b = 2B and c = B2 + C

which gives B =

b

2

and C = c – B2

Using these two results we can now set about completing the square in some simple cases.

Example 1

Write each of the following expressions in the form (x + B)2 + C.

(a)

x2 + 6x + 1

(b)

x2 + 4x – 2

(c)

x2 + 2x

Example 2

Solve the following equations by completing the square.

(a)

x2 – 4x – 5 = 0

(b)

x2 + 6x – 1 = 0