Question

If α and β are the roots of the equation x2 - 2x + 3 = 0, then the equation whose roots are α + β-1 and β + α-1 is
1. x² + 8x + 16 = 0
2. 3x² - 8x + 16 = 0
3. 3x² + 8x - 16 = 0
4. None of these

Answer 1

Correct Answer - Option 2 : 3x² - 8x + 16 = 0

Concept:

General Form of Quadratic Equation, ax² + bx + c = 0 

1. Sum of roots, α + β = \(\rm \frac{-b}{a}\)
2. Products of roots, αβ = \(\rm \frac{c}{a}\)
3. The quadratic equation whose roots are α and β is given by x² - (α + β)x + αβ = 0

 

Calculation:

Given

⇒ x2 - 2x + 3 = 0

⇒ sum of root α + β = \(\rm \frac{-b}{a}\) = \(\rm \frac{-(-2)}{1}\)  = 2

⇒ α + β = 2      ....(1)

⇒ αβ = \(\rm \frac{c}{a} = \frac{3}{1}\) = 3

⇒ αβ = 3      ....(2)

 

Given two roots are α + β-1 and β + α-1 

Sum of roots =  α + β-1 + β + α-1 

=  α + β +  β-1  + α-1 

=  α + β + (α + β) α-1 β-1 

= 2 + \(\rm \frac{2}{3}\)

= \(\rm \frac{8}{3}\)

 

Products of roots =  (α + β-1 )  (β + α-1)

= αβ + 1 + 1 + α-1 β-1

= 3 + 2 + \(\rm \frac{1}{3}\)

= \(\rm \frac{16}{3}\)

 

The quadratic equation is 

⇒  x2 - x (Sum of roots) + Products of roots = 0

⇒ x2 - x (\(\rm \frac{8}{3}\)) + \(\rm \frac{16}{3}\) = 0

⇒ 3x2 - 8x + 16 = 0

Relation between Roots and Coefficients

• If the roots of quadratic equation ax² + bx + c, a ≠ 0, are α and ß,then α + β = \(\rm \frac{-b}{a} = - \frac{Coefficient of\: x}{Coefficient of\: x^{2}}\) and αβ = \(\rm \frac{c}{a} = \frac{Constant term}{Coefficient of\: x^{2}}\)
• The quadratic equation whose roots are α and β is given by x2 - (α + β)x + αβ = 0