Question
Download Solution PDFLet α and β (α > β) be the roots of the equation x2 - 8x + q = 0. If α2 - β2 = 16, then what is the value of q?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
The Standard Form of a Quadratic Equation is ax2 + bx + c = 0
Sum of roots = −b/a
Product of roots = c/a
Formulae
(α - β)2 + 4.α.β = (α + β)2
Calculation:
Given:
x2 - 8x + q = 0 and α2 - β2 = 16
Sum of roots = α + β = −b/a = -(- 8) = 8 ------(i)
Product of roots = α.β = q ------(ii)
We have α2 - β2 = 16
⇒ (α + β)(α - β) = 16
⇒ 8 × (α - β) = 16
⇒ (α - β) = 2 ------(iii)
We know that, (α - β)2 + 4.α.β = (α + β)2
Putting the value of equation (i), (ii) and (iii), we get,
⇒ 22 + 4q = 82
⇒ 4q = 64 - 4 = 60
⇒ q = 15
∴ The value of q is 15
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