# If α and β are roots of $\rm 2x^2+5x-3=0$, then |α - β| is equal to

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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. $\frac{7}{4}$

## 1 Answer

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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.

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