5x² +6x-2=0
Discriminant = b² - 4ac
Here, a = 5, b = (- 6), c = (- 2)
Then, b² - 4ac = (- 6)² - 4 × 5 × (- 2)
= 36 + 40 = 76 > 0
So the equation has real roots and two distinct roots,
Again, 5x² - 5x = 2 (dividing both sides by 5)
⇒ x² - 6/5x + 9/25 = 2/5 + 9/5
On adding square of half of coefficient of x
⇒ x² - 6/5x + 9/25 = 2/5 + 9/25
⇒ x - 3/5 = ± √19/5
⇒ x = 3 + √19/5 or 3 - √19/5
Verification :-
= 5[3 + √19/5]² - 6[3 + √19/5] - 2
= 9 + 6√19 + 19/5 - (18 + 6√19/5) - 2
= 28 + 6√19/5 - 18 + 6√19/5 - 2
= 28 + 6√19 - 18 - 6√19 - 10/5 = 0
Similarly,
5[3 + √19/5]² - 6[3 + √19/5] - 2 = 0
Hence Verified.
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Answers & Comments
5x² +6x-2=0
Discriminant = b² - 4ac
Here, a = 5, b = (- 6), c = (- 2)
Then, b² - 4ac = (- 6)² - 4 × 5 × (- 2)
= 36 + 40 = 76 > 0
So the equation has real roots and two distinct roots,
Again, 5x² - 5x = 2 (dividing both sides by 5)
⇒ x² - 6/5x + 9/25 = 2/5 + 9/5
On adding square of half of coefficient of x
⇒ x² - 6/5x + 9/25 = 2/5 + 9/25
⇒ x - 3/5 = ± √19/5
⇒ x = 3 + √19/5 or 3 - √19/5
Verification :-
= 5[3 + √19/5]² - 6[3 + √19/5] - 2
= 9 + 6√19 + 19/5 - (18 + 6√19/5) - 2
= 28 + 6√19/5 - 18 + 6√19/5 - 2
= 28 + 6√19 - 18 - 6√19 - 10/5 = 0
Similarly,
5[3 + √19/5]² - 6[3 + √19/5] - 2 = 0
Hence Verified.