Question 5: 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.

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-5 is a root of 2x^{2} + Px - 15 = 0

Because -5 is a root of the equation, substituting x = -5 in the equation will satisfy the equation.

Substitute x = -5 in the equation to find P.

2(-5)^{2} + P(-5) - 15 = 0

50 - 5P - 15 = 0

5P = 35 or P = 7

The equation P(x^{2} + x) + K = 0 has equal roots.

i.e., 7x^{2} + 7x + K = 0 has equal roots

If a quadratic equation ax^{2} + bx + c = 0 has real and equal roots, its discriminant, b^{2} - 4ac = 0

i.e., 7^{2} - 4 × 7 × K = 0

28K = 49 or K = \\frac{49}{28}) = \\frac{7}{4})

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