Question

If α and β are roots of \(\rm 2x^2+5x-3=0\), then |α - β| is equal to
1. 2√3
2. \(\frac{5}{3}\)
3. \(\frac{7}{2}\)
4. \(\frac{7}{4}\)

Answer 1

Correct Answer - Option 3 : \(\frac{7}{2}\)

Concept:

Consider a quadratic equation: ax² + bx + c = 0.

Let, α and β are the roots.

• Sum of roots = α + β = -b/a
• Product of the roots = αβ = c/a

\(\rm (a+b)^2=(a-b)^2+4ab\)

 

Calculation:

Given equation: \(\rm 2x^2+5x-3=0\)

α and β are the roots of the given equation.

∴ Sum of roots = α + β = \(-\frac{5}{2}\) and 

Product of the roots = αβ = \(\frac{-3}{2}\)

Now, we know, \(\rm (a+b)^2=(a-b)^2+4ab\)

∴ (α + β)² = (α - β)² + 4αβ 

⇒ \(\rm (-\frac52)^2=(\alpha -\beta)^2+4(-\frac32)\)

\(\Rightarrow \rm (\alpha -\beta)^2=\frac{25}{4}+6\)

\(=\frac{49}{4}\)

\(\Rightarrow \rm |(\alpha -\beta)|=\frac{7}{2}\)

Hence, option (3) is correct.