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# If −5 is a Root of the Quadratic Equation 2 X 2 + P X − 15 = 0 and the Quadratic Equation P ( X 2 + X ) + K = 0 Has Equal Roots, Find the Value of K. - CBSE Class 10 - Mathematics

ConceptSolutions of Quadratic Equations by Factorization

#### Question

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

#### Solution

The given quadratic equation is $2 x^2 + px - 15 = 0$ and one root is −5.
Then, it satisfies the given equation.

$2 \left( - 5 \right)^2 + p\left( - 5 \right) - 15 = 0$

$\Rightarrow 50 - 5p - 15 = 0$

$\Rightarrow - 5p = - 35$

$\Rightarrow p = 7$

The quadratic equation $p( x^2 + x) + k = 0$, has equal roots.

Putting the value of p, we get

$7\left( x^2 + x \right) + k = 0$

$\Rightarrow 7 x^2 + 7x + k = 0$

Here,

$a = 7, b = 7 \text { and } c = k$.

As we know that $D = b^2 - 4ac$

Putting the values of  $a = 7, b = 7 \text { and } c = k$

$D = \left( 7 \right)^2 - 4\left( 7 \right)\left( k \right)$

$= 49 - 28k$

The given equation will have real and equal roots, if D = 0

Thus,

$49 - 28k = 0$

$\Rightarrow 28k = 49$

$\Rightarrow k = \frac{49}{28}$

$\Rightarrow k = \frac{7}{4}$

Therefore, the value of k is  $\frac{7}{4}$.

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#### APPEARS IN

Solution If −5 is a Root of the Quadratic Equation 2 X 2 + P X − 15 = 0 and the Quadratic Equation P ( X 2 + X ) + K = 0 Has Equal Roots, Find the Value of K. Concept: Solutions of Quadratic Equations by Factorization.
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