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

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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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Solution

Given –5 is a root of the quadratic equation 2x2 + px – 15 = 0.

∴-5 satisfies the given equation.

∴ 2(5)2+ p(-5)-15 = 0

∴ 50 - 5p - 15= 0

∴ 35-5p = 0

∴5p = 35 ⇒ p = 7

Substituting p = 7 in p(x2 + x)+ k= 0,we get

7(x2+x)+k=0

∴7x2 + 7x + k = 0

The roots of the equation are equal.

∴ Discriminant b2 - 4ac = 0

Here, a=7, b=7, c=k

b2-4ac=0

∴ (7)2 - 4(7)(k)=0

∴ 49-28k= 0

∴28k =  49

∴ k = `49/28=7/4`

Concept: Nature of Roots
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