Question

$4x {}^{2} - x + 4 \\ \: \: find \: the \: value \: of \: \: \: 1 \alpha + 1 \beta$
​

Answer 1

Step-by-step explanation:

4x^2 -x +4 = 0

let's roots be alpha and beta

we know sum of roots = -b/a

alpha + beta = 1/4

Answer 2

${\bold{\underline{\underline{Answer:}}}}$

${\bold{\therefore \alpha+\beta=\frac{1}{4}}}$

${\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

• In the given question information given about a quadratic eqn whose roots are alpha and beta.

• We have to find the given value.

$\underline \bold{Given : } \\ \implies \alpha \: and \: \beta \: \in ({4x}^{2} - x + 4) \\ \\ \underline \bold{To \: Find : } \\ \implies \alpha + \beta = ?$

• According to given question :

$\bold{By \: Quadratic \: eqn : } \\ \implies 4 {x}^{2} - x + 4 = 0 \\ \\ \implies x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac } }{2a} \\ \\ \implies x = \frac{ - ( - 1) \pm \sqrt{ {( - 1)}^{2} - 4 \times 4 \times 4 } }{2 \times 4} \\ \\ \implies x = \frac{1 \pm \sqrt{1 - 16} }{2\times 4}$

$\implies x = \frac{1 \pm \sqrt{ - 15} }{8} \\ \\ \bold{note = \sqrt{ - 15} = i} \\ \\ \implies x = \frac{1 \pm i}{8} \\ \\ \bold{\implies \alpha = \frac{1 + i}{8} }\\ \\ \bold{ \implies \beta = \frac{1 - i}{8} } \\ \\ \bold{for \: finding \: values : } \\ \implies \alpha + \beta = \frac{1 + i}{8} + \frac{1 - i}{8} \\ \\ \implies \alpha + \beta = \frac{1 \cancel{+ i }+ 1 \cancel{ - i}}{8} \\ \\ \implies \alpha + \beta = \frac{\cancel2}{\cancel8} \\ \\ \bold{\implies \alpha + \beta = \frac{1}{4} }$