Check whether the equation 5x2
– 6x – 2 = 0 has real roots and if it has, find them by
the method of completing the square. Also verify that roots obtained satisfy the given
equation.
Answers
5x² - 6x - 2 = 0
Check if the quadratic equation has real roots:
b² - 4ac = (-6)² - 4(5)(-2)
b² - 4ac =36 + 40
b² - 4ac = 76
b² - 4ac > 0
⇒ The quadratic equation has real roots
Find the roots by completing the square:
5x² - 6x - 2 = 0
(Divide by 5 through to get the coefficient of x² as 1)
x² - 6/5 x - 2/5 = 0
(Add 2/5 to both sides)
x² - 6/5 x = 2/5
(Add 1/2 the coefficient of x to both sides )
x² - 6/5 x + (6/10)² = 2/5 + (6/10)²
(Complete the square)
(x - 6/10)² = 2/5 + 36/100
(Simplify)
(x - 3/5)² = 19/25
(Square root both sides)
x - 3/5 = ±√(19/25)
(Evaluate x)
x = √(19/25) + 3/5 or x = -√(19/25)+ 3/5
x = 1.472 or 0.272
Verify:
Both the value of x are positive
⇒ The quadratic equation has real roots
Answer: x = 1.472 or 0.272
Given:
5x² - 6x - 2 = 0
To find:
Whether they have real roots.
If they have, find them by the method of completing the square.
To verify that roots obtained satisfy the given equation.
Solution:
To check if the quadratic equation has real roots,
b² - 4ac > 0
Here,
a = 5
b = -6
c = -2
Substituting,
(-6)² - 4(5)(-2)
76
b² - 4ac > 0
Hence, the quadratic equation has real roots.
To find the roots by completing the square,
Dividing the equation by 5,
x² - 6/5 x - 2/5 = 0
Adding 2/5 on both sides,
x² - 6/5 x = 2/5
Adding 1/2 the coefficient of x on both sides,
x² - 6/5 x + ( 6 / 10 ) ^2 = 2/5 + ( 6 / 10 )^2
( x - 6 / 10 )^2 = 2/5 + 36/100
Solving,
( x - 3 / 5 )^2 = 19/25
x - 3/5 = ± √ ( 19 / 25 )
Finding x,
x = √ ( 19 / 25 ) + 3 / 5
x = - √ ( 19 / 25 ) + 3 / 5
Hence,
x = 1.472
x = 0.272
To verify that roots obtained satisfy the given equation,
Positive roots.
As the roots are positive, the equation has real roots.