Question

1f alpha and beta are the zeros of the quadratic polynomial p(x) = 4x²- 5x -1, find the value of a²ß+ aß²

Answer 1

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

$\green{\tt{\therefore{\alpha^{2}\beta+\alpha\beta^{2}=\frac{-5}{16}}}}$

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

$\green{\underline \bold{Given :}} \\ \tt: \implies p(x) = 4 {x}^{2} - 5x - 1 \\ \\ \red{\underline \bold{To \: Find:}} \\ \tt: \implies { \alpha }^{2} \beta + \alpha { \beta }^{2} = ?$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies {4x}^{2} - 5x - 1 = 0 \\ \\ \tt \circ \: a = 4 \\ \\ \tt \circ \: b = - 5 \\ \\ \tt \circ \ c = - 1 \\ \\ \tt:\implies sum\:of\:zeroes=\frac{-b}{a}\\\\ \tt:\implies \alpha+\beta=\frac{-(-5)}{4}=\frac{5}{4}\\\\ \tt:\implies Product\:of\:zeroes=\frac{c}{a}\\\\ \tt:\implies \alpha\beta=\frac{-1}{4}\\\\ \bold{For\:finding\:value}\\\tt:\implies \alpha^{2}\beta+\alpha\beta^{2}\\\\ \tt:\implies \alpha\beta{(\alpha+\beta)}\\\\ \tt:\implies \frac{-1}{4}\times \frac{5}{4}\\\\ \green{\tt:\implies \frac{-5}{16}}\\\\ \green{\tt\therefore \alpha^{2}\beta+\alpha\beta^{2}=\frac{-5}{16}}$

$\bold{Alternate \:method} \\ \tt: \implies {4x}^{2} - 5x - 1 = 0 \\ \\ \tt \circ \: a = 4 \\ \\ \tt \circ \: b = - 5 \\ \\ \tt \circ \: c = - 1 \\ \\ \tt: \implies x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac } }{2a} \\ \\ \tt: \implies x = \frac{ - ( - 5) \pm \sqrt{( - 5)^{2} - 4 \times 4 \times ( - 1) } }{2 \times 4} \\ \\ \tt: \implies x = \frac{5 \pm \sqrt{25 + 16} }{8} \\ \\ \green{\tt: \implies x = \frac{5 \pm \sqrt{41} }{8}} \\ \\ \bold{for \: finding \: value} \\ \tt: \implies { \alpha }^{2} \beta + \alpha { \beta }^{2} \\ \\ \tt: \implies (\frac{5 + \sqrt{41} }{8} )^{2} \times \frac{5 - \sqrt{41} }{8} + \frac{5 + \sqrt{41} }{8} \times ( { \frac{5 - \sqrt{41} }{8} })^{2} \\ \\ \tt \circ \: \sqrt{41} = 6.4\\ \\ \tt: \implies (\frac{5 + 6.4}{8} )^{2} \times \frac{5 - 6.4}{8} + \frac{5 + 6.4}{8} \times {( \frac{5 - 6.4}{8} })^{2} \\ \\ \tt: \implies (\frac{11.4}{8} )^{2} \times \frac{ - 1.4}{8} + \frac{11.4}{8} \times (\frac{ - 1.4}{8} )^{2} \\ \\ \tt: \implies \frac{129.96}{64} \times \frac{ - 1.4}{8} + \frac{11.4}{8} \times \frac{ - 1.96}{64} \\ \\ \tt: \implies \frac{ - 181.496}{512} + \frac{ - 22.344}{512} \\ \\ \tt: \implies \frac{181.496 - 22.344}{ 512} \\ \\ \tt: \implies \frac{ - 203.84}{512} \\ \\ \green{\tt: \implies - 0.39} \\ \\ \green{\tt\therefore { \alpha }^{2} \beta + { \alpha \beta }^{2} \approx - 0.4}$

Answer 2

Answer:

Answer: -5/16

Explanation:-

Step-by-step explanation:

Given,

p(x) = 4x² - 5x - 1

On comparing this polynomial with standard form of Polynomial of second degree,

p(x) = ax² + bx + c

We get,

a = 4

b = - 5

c = - 1

We know that,

Sun of Roots = -b/a

α + β = -(-5)/4

α + β = 5/4

And,

Product of Roots = c/a

αβ = -1/4

Now,

α²β + αβ²

= αβ(α + β)

= -1/4( 5/4 )

= -5/16