Question

if a and b are zeroes of 2x²+5x+k and a²+b²+ab=21/4,find a and b.
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Answer 1

Answer:

$a = - 2 \: and \: b = - \frac{1}{2}$

Step-by-step explanation:

if a and b are roots of the quadratic equation, then

$a + b = - \frac{5}{2} \\ ab = \frac{k}{2}$

Niw, using

$(a + b)^{2} = {a}^{2} + {b}^{2} + 2ab$

we can get

$( - \frac{5}{2} )^{2} = {a}^{2} + {b}^{2} + 2( \frac{k}{2} ) \\ \rightarrow \: {a}^{2} + {b}^{2} = \frac{25}{4} - k$

Using it in second equation (a and b equation), we can get

$\frac{25}{4} - k + \frac{k}{2} = \frac{21}{4} \\ \rightarrow \: k = 2$

This means the quadratic equation would be

$2 {x}^{2} + 5x + 2 = 0$

To calculate roots of this

$2 {x}^{2} + 4x + x + 2 = 0 \\ 2x(x + 2) + 1(x + 2) = 0 \\ (2x + 1)(x + 2) = 0 \\ x = - 2 \: or \: - \frac{ 1}{2}$

So,

$a = - 2 \: and \: b = - \frac{1}{2}$