Check whether the equation 6x² - 7x + 2 = 0 has real roots, and if it has, find them by the method of completing the squares
Solution:
Given, the equation is 6x² - 7x + 2 = 0
We have to find the roots of the equation by the method of completing the squares.
A quadratic equation ax² + bx + c = 0 has 2 distinct real roots when the discriminant of the equation is greater than zero.
Here, a = 6, b = -7 and c = 2
Discriminant = b² - 4ac
= (-7)² - 4(6)(2)
= 49 - 48
= 1 > 0
So, the equation has 2 distinct real roots.
Now finding the roots by completing the squares method,
Multiplying the equation by 6,
36x² - 42x + 12 = 0 ----------------------- (1)
By using algebraic identity,
(a - b)² = a² - 2ab + b² -------------------- (2)
Comparing (1) and (2),
a² = 36
a = 6
2ab = 42
2(6)b = 42
12b = 42
b = 42/12
b = 14/4
b = 7/2
b² = (7/2)² = 49/4
Now, 36x² - 42x + 12 + 49/4 - 49/4 = 0
By grouping,
36x² - 42x + 49/4 + 12 - 49/4 = 0
(6x - 7/2)² + 12 - 49/4 = 0
(6x - 7/2)² + (48 - 49)/4 = 0
(6x - 7/2)² - 1/4 = 0
(6x - 7/2)² = (1/2)²
Taking square root,
6x - 7/2 = ±1/2
Now, 6x - 7/2 = 1/2
6x = 1/2 + 7/2
6x = 8/2
6x = 4
x = 4/6
x = 2/3
Also, 6x - 7/2 = -1/2
6x = 7/2 - 1/2
6x = 6/2
6x = 3
x = 3/6
x = 1/2
Therefore, the roots of the equation are 2/3 and 1/2.
✦ Try This: Check whether the equation 5x² - 6x + 2 = 0 has real roots, and if it has, find them by the method of completing the squares
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 4
NCERT Exemplar Class 10 Maths Exercise 4.4 Sample Problem 1
Check whether the equation 6x² - 7x + 2 = 0 has real roots, and if it has, find them by the method of completing the squares
Summary:
The equation 6x² - 7x + 2 = 0 has real roots. The roots of the equation by the method of completing the squares are 2/3 and 1/2.
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