Question

if -5 is a root of the quadratic equation 2x² + px - 15 = 0 and the quadratic equation p(x²+ x) + k =0 has equal roots find the value of k​

Answer 1

Step-by-step explanation:

If the roots are equal(double root) it means that discriminant of quadratic equation ... For k=6 or k= -6 given equation has real and equal roots ...

Answer 2

$\large\underline\blue{\bold{Given \:  Question :-  }}$

If -5 is a root of the quadratic equation 2x² + px - 15 = 0 and the quadratic equation p(x²+ x) + k =0 has equal roots find the value of k.

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$\huge{AηsωeR} ✍$

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$\begin{gathered}\begin{gathered}\bf Given - \begin{cases} &\sf{ -5 \: is \: a \: root \: of \: 2x² + px - 15 = 0} \\ &\sf{p(x²+ x) + k =0 \: has \: equal \: roots} \end{cases}\end{gathered}\end{gathered}$

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$\begin{gathered}\begin{gathered}\bf Find - \begin{cases} &\sf{the \: value \: of \: k} \end{cases}\end{gathered}\end{gathered}$

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$\large\underline\purple{\bold{Solution :- }}$

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$\large\underline\red{\bold{❥︎Step :- 1 }}$

☆ Since -5 is a root of the quadratic equation 2x² + px - 15 = 0

$\sf \:  ⟼ \therefore \: 2 \times {( - 5)}^{2} - 5p - 15 = 0$

$\sf \:  ⟼50 - 5p - 15 = 0$

$\sf \:  ⟼35 - 5p = 0$

$\bf\implies \: \: p \: = \: 7$

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$\large\underline\red{\bold{❥︎Step :- 2 }}$

☆ The quadratic equation p(x²+ x) + k =0 has equal roots.

The equation can be rewritten as

$\sf \:  ⟼p {x}^{2} + px + k = 0 \:$

☆ Since, equation has equal roots.

☆ Therefore, Discriminant = 0

$\begin{gathered}\begin{gathered}\bf here - \begin{cases} &\sf{a = p} \\ &\sf{b = p}\\ &\sf{c = k} \end{cases}\end{gathered}\end{gathered}$

☆ Now, as we know, Discriminant = 0.

$\bf\implies \: \: {b}^{2} - 4ac = 0$

$\sf \:  ⟼ \: {p}^{2} - 4pk = 0$

$\sf \:  ⟼ \: p(p - 4k) = 0$

$\sf \:  ⟼7(7 - 4k) = 0 \: \: \: ( \because \: p \: = \: 7)$

$\bf\implies \:k = \dfrac{7}{4}$

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