Maths

QUADRATIC EQUATIONS 81
Therefore, the solutions of the equations are
Let us verify our solutions.
3
x and 1.
2
3
3 3
Putting x in 2x
2
2 ✁ ✂ ✂ ✁
– 5x + 3 = 0, we get 2 ✆ ✝ ✆ ✝ –5 ✄ 3 ☎ 0, which is
2 ✞ ✟ ✞ ✟ 2
correct. Similarly, you can verify that x = 1 also satisfies the given equation.
In Example 7, we divided the equation 2x 2 – 5x + 3 = 0 throughout by 2 to get
x 2 – 5 x 3 ✠ = 0 to make the first term a perfect square and then completed the
2 2
square. Instead, we can multiply throughout by 2 to make the first term as 4x 2 = (2x) 2
and then complete the square.
This method is illustrated in the next example.
Example 8 : Find the roots of the equation 5x 2 – 6x – 2 = 0 by the method of completing
the square.
Solution : Multiplying the equation throughout by 5, we get
This is the same as
25x 2 – 30x – 10 = 0
(5x) 2 – 2 × (5x) × 3 + 3 2 – 3 2 – 10 = 0
i.e., (5x – 3) 2 – 9 – 10 = 0
i.e., (5x – 3) 2 – 19 = 0
i.e., (5x – 3) 2 =19
i.e., 5x – 3 = 19 ✡
i.e., 5x = 3 19 ✡
So, x = 3 19 ☛
5
2
Therefore, the roots are 3 19 ☞
5
Verify that the roots are 3 19 ☞
5
and 3 19 ✌
5
and 3 19 ✌
5
.
.