106 Basic Engineering Mathematics
Adding to both sides (half the coefficient of x)
2 gives
x
2
+
9
2
x +
9
4
2
=
9
4
2
− 4
The LHS is now a perfect square. Thus,
x +
9
4
2
=
81
16
− 4 =
81
16
−
64
16
=
17
16
Taking the square root of both sides gives
x +
9
4
=
17
16
= ±1.031
Hence,
x = −
9
4
± 1.031
i.e. x = −1.22 or −3.28, correct to 3 significant figures.
Problem 17. By completing the square, solve the
quadratic equation 4.6y 2 + 3.5y − 1.75 = 0, correct
to 3 decimal places
4.6y
2
+ 3.5y − 1.75 = 0
Making the coefficient of y 2 unity gives
y
2
+
3.5
4.6
y −
1.75
4.6
= 0
and rearranging gives
y
2
+
3.5
4.6
y =
1.75
4.6
Adding to both sides (half the coefficient of y) 2 gives
y
2
+
3.5
4.6
y +
3.5
9.2
2
=
1.75
4.6
+
3.5
9.2
2
The LHS is now a perfect square. Thus,
y +
3.5
9.2
2
= 0.5251654
Taking the square root of both sides gives
y +
3.5
9.2
=
√
0.5251654 = ±0.7246830
Hence,
y = −
3.5
9.2
± 0.7246830
i.e.
y = 0.344 or −1.105
Now try the following Practice Exercise
Practice Exercise 55 Solving quadratic
equations by completing the square
(answers on page 346)
Solve the following equations correct to 3 decimal
places by completing the square.
1. x 2 + 4x + 1 = 0
2. 2x 2 + 5x − 4 = 0
3. 3x 2 − x − 5 = 0
4. 5x 2 − 8x + 2 = 0
5. 4x 2 − 11x + 3 = 0
6. 2x 2 + 5x = 2
14.4 Solution of quadratic equations
by formula
Let the general form of a quadratic equation be given
by ax 2 + bx + c = 0, where a, b and c are constants.
Dividing ax
2
+ bx + c = 0 by a gives
x
2
+
b
a
x +
c
a
= 0
Rearranging gives
x
2
+
b
a
x = −
c
a
Adding to each side of the equation the square of half
the coefficient of the term in x to make the LHS a perfect
square gives
x
2
+
b
a
x +
b
2a
2
=
b
2a
2
−
c
a
Rearranging gives
x +
b
a
2
=
b
2
4a 2 −
c
a
=
b
2
− 4ac
4a 2
Taking the square root of both sides gives
x +
b
2a
=
b 2 − 4ac
4a 2
=
±
√
b 2 − 4ac
2a
Hence,
x = −
b
2a
±
√
b 2 − 4ac
2a
i.e. the quadratic formula is
x =
−b ±
√
b 2 − 4ac
2a
(This method of obtaining the formula is completing the
square − as shown in the previous section.)
In summary,
if ax
2
+ bx + c = 0 then x =
−b ±
√
b 2 − 4ac
2a
This is known as the quadratic formula.
Adding to both sides (half the coefficient of x)
2 gives
x
2
+
9
2
x +
9
4
2
=
9
4
2
− 4
The LHS is now a perfect square. Thus,
x +
9
4
2
=
81
16
− 4 =
81
16
−
64
16
=
17
16
Taking the square root of both sides gives
x +
9
4
=
17
16
= ±1.031
Hence,
x = −
9
4
± 1.031
i.e. x = −1.22 or −3.28, correct to 3 significant figures.
Problem 17. By completing the square, solve the
quadratic equation 4.6y 2 + 3.5y − 1.75 = 0, correct
to 3 decimal places
4.6y
2
+ 3.5y − 1.75 = 0
Making the coefficient of y 2 unity gives
y
2
+
3.5
4.6
y −
1.75
4.6
= 0
and rearranging gives
y
2
+
3.5
4.6
y =
1.75
4.6
Adding to both sides (half the coefficient of y) 2 gives
y
2
+
3.5
4.6
y +
3.5
9.2
2
=
1.75
4.6
+
3.5
9.2
2
The LHS is now a perfect square. Thus,
y +
3.5
9.2
2
= 0.5251654
Taking the square root of both sides gives
y +
3.5
9.2
=
√
0.5251654 = ±0.7246830
Hence,
y = −
3.5
9.2
± 0.7246830
i.e.
y = 0.344 or −1.105
Now try the following Practice Exercise
Practice Exercise 55 Solving quadratic
equations by completing the square
(answers on page 346)
Solve the following equations correct to 3 decimal
places by completing the square.
1. x 2 + 4x + 1 = 0
2. 2x 2 + 5x − 4 = 0
3. 3x 2 − x − 5 = 0
4. 5x 2 − 8x + 2 = 0
5. 4x 2 − 11x + 3 = 0
6. 2x 2 + 5x = 2
14.4 Solution of quadratic equations
by formula
Let the general form of a quadratic equation be given
by ax 2 + bx + c = 0, where a, b and c are constants.
Dividing ax
2
+ bx + c = 0 by a gives
x
2
+
b
a
x +
c
a
= 0
Rearranging gives
x
2
+
b
a
x = −
c
a
Adding to each side of the equation the square of half
the coefficient of the term in x to make the LHS a perfect
square gives
x
2
+
b
a
x +
b
2a
2
=
b
2a
2
−
c
a
Rearranging gives
x +
b
a
2
=
b
2
4a 2 −
c
a
=
b
2
− 4ac
4a 2
Taking the square root of both sides gives
x +
b
2a
=
b 2 − 4ac
4a 2
=
±
√
b 2 − 4ac
2a
Hence,
x = −
b
2a
±
√
b 2 − 4ac
2a
i.e. the quadratic formula is
x =
−b ±
√
b 2 − 4ac
2a
(This method of obtaining the formula is completing the
square − as shown in the previous section.)
In summary,
if ax
2
+ bx + c = 0 then x =
−b ±
√
b 2 − 4ac
2a
This is known as the quadratic formula.