Page 22 - CHAPTER 4 (Quadratic equations)

CHAPTER 4

QUADRATIC EQUATIONS

3
Example 3: Find the roots of 4x +3x+5=0 by the method of
completing the square.

2
Solution: Note that 4x -3x-5=0 is the same as

2
2
3
3
3
2
(2x) +2×(2x)× + ( ) - ( ) +5=0
4 4 4
2
3
i.e., (2x+ ) - 9 +5=0
4 16
2
3
i.e., (2x+ ) + 71 =0
4 16
2
3
i.e., (2x+ ) - -71 <0
4 6

2
3
but (2x+ ) cannot be negative for any real value of x(why?) . So, there
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is no real value of x satisfying the given equation. Therefore , the given

equation has real roots.

Now you have seen several examples of the use of the method of

completing the square . So, let us give this method in general.

Consider the quadratic equation + − = 0 ( ≠ 0) . dividing
2
throughout by a, we get

b c
2
x + x+ =0
a a
2
2
b
c
b
This is the same as (x+ ) - ( ) + =0
2a 2a a
2
2
b
i.e., (x+ ) - b -4ac =0
2a 4a 2
So the roots of the given equation are the same as those of
2
2
2
2
b
b
(x+ ) - b -4ac =0, i.e., those of (x+ ) = b -4ac
2a 4a 2 2a 4a 2
2
If b -4ac≥0, then by taking the square roots in (1), we get
2
√
b ± b -4ac
x+ =
2a 2a

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