Question

$\sf\small\underline \red{Question:-}$
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Answer 1

Solution

Given :-

• Polynomial equation, x² - 2x - k = 0,
• p & q be zeroes
• p² + q² = 34__________________(1)

Find :-

• Value of k.

Explantion

Using Formula

★ sum of zeroes = -(coefficient of x)/(coefficient of x²)

★ product of zeroes = (constant part)/(coefficient of x)

So

==> p + q = -(-2)/1

==> p + q = 2_______________(2)

And

==> p.q = -k ______________(3)

Squaring both side of equ(2)

==> (p + q )² = 2²

Using Formula

★(a + b)² = a² + b² + 2ab

==> p² + q² + 2pq = 4

keep value by equ(1)

==> 34 + 2pq = 4

==> 2pq = 4 - 34

==> 2pq = -30

==> pq = -30/2

==> pq = -15__________________(4)

Compare equ(3) & equ(4)

==> k = -15

Hence

• Value of k will be k = -15

___________________

Answer 2

Answer:

Given, p and q are the zeros of the quadratic polynomial

${x}^{2} - 2x - k$

and

${p}^{2} + {q}^{2} = 34$

Value of k = to find

$now \: \\ \sf \: sum \: of \: zeros \: \: \: p + q = - \frac{ - 2}{1} = 2 \\ \\ \sf \: product \: of \: zeros \: \: pq = \frac{ - k}{1} = - k \\ \\ \: we \: know \: that \\ \\ \sf \: \: \: \: \: \: \: {p}^{2} + {q}^{2} = {(p + q)}^{2} - 2pq \\ \sf \: \implies \: 34 = {2}^{2} - 2 \times ( - k) \\ \sf \: \implies \: 34 = 4 + 2k \\ \sf \: \implies \: 2k = 34 - 4 = 30 \\ \sf \: \implies \: k = \frac{30}{2} \\ \sf \: \implies \: k = 15$