Question

If α and β are the roots of quadratic equation x2+5x+a = 0 and
2 α+5β= -1 then a is equal to

Answer 1

GIVEN :–

• A quadratic equation x² + 5x + a = 0 have two roots α and β.

• Relation in roots is 2α + 5β = -1 .

TO FIND :–

• Value of 'a' = ?

SOLUTION :–

• Let's find roots of quadratic equation –

$\\ \implies{ \bold{ {x}^{2} + 5x + a = 0 }}$

• We know that –

$\\ \to \: \: \: \: { \bold{ x = \dfrac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} }}$

• So that –

$\\ \implies \: { \bold{ x = \dfrac{ - (5) \pm \sqrt{ {5}^{2} - 4(1)a} }{2(1)} }}$

$\\ \implies \: { \bold{ x = \dfrac{ - 5 \pm \sqrt{ 25 - 4a} }{2} }}$

• Then roots –

$\\ \implies \: { \bold{ \alpha = \dfrac{ - 5 + \sqrt{ 25 - 4a} }{2} \: \: , \: \: \beta = \dfrac{ - 5 - \sqrt{25 - 4a} }{2} }}$

• Now According to the given condition –

$\\ \implies \: { \bold{ 2 \alpha + 5 \beta = - 1}}$

• Put the values of α and β –

$\\ \implies \: { \bold{ 2 \left( \frac{ - 5 + \sqrt{25 - 4a} }{2} \right) + 5 \left( \frac{ - 5 - \sqrt{25 - 4a} }{2} \right) = - 1}}$

$\\ \implies \: { \bold{ 2 \left( \frac{ - 5 + \sqrt{25 - 4a} }{2} + \frac{ - 5 - \sqrt{25 - 4a} }{2} \right) + 3 \left( \frac{ - 5 - \sqrt{25 - 4a} }{2} \right) = - 1}}$

$\\ \implies \: { \bold{ 2 \left( \frac{ - 5 }{2} + \frac{ - 5 }{2} \right) + 3 \left( \frac{ - 5 - \sqrt{25 - 4a} }{2} \right) = - 1}}$

$\\ \implies \: { \bold{ \cancel2 \left( \frac{ - 10 }{ \cancel2} \right) + 3 \left( \frac{ - 5 - \sqrt{25 - 4a} }{2} \right) = - 1}}$

$\\ \implies \: { \bold{ - 10+ 3 \left( \frac{ - 5 - \sqrt{25 - 4a} }{2} \right) = - 1}}$

$\\ \implies \: { \bold{ 3 \left( \frac{ - 5 - \sqrt{25 - 4a} }{2} \right) = 10 - 1}}$

$\\ \implies \: { \bold{ \cancel3 \left( \frac{ - 5 - \sqrt{25 - 4a} }{2} \right) = \cancel 9}}$

$\\ \implies \: { \bold{ \left( \frac{ - 5 - \sqrt{25 - 4a} }{2} \right) = 3}}$

$\\ \implies \: { \bold{ { - 5 - \sqrt{25 - 4a} } = 6}}$

$\\ \implies \: { \bold{ { - \sqrt{25 - 4a} } = 6 + 5}}$

$\\ \implies \: { \bold{ { - \sqrt{25 - 4a} } = 11}}$

• Square on both side –

$\\ \implies \: { \bold{ { (- \sqrt{25 - 4a} \: ) {}^{2} } = (11)^{2} }}$

$\\ \implies \: { \bold{ 25 - 4a = 121 }}$

$\\ \implies \: { \bold{ - 4a = 121 - 25 }}$

$\\ \implies \: { \bold{ 4a = - 96 }}$

$\\ \implies \: { \bold{ a = - \cancel \dfrac{96}{4} }}$

$\\ \implies \: { \bold{ a = -24 }}$

Hence , Value of a is -24 .