Question

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

equation p(x2+x)+k=0 and has equal roots, find the value of k.​

Answer 1

Question:—

1. If -5 is a root of the quadratic equation 2x²+px-15=0 , find the value of p.
2. The quadratic equation px²+x+k=0 and has equal roots, find the value of k.

Solution:—

(1) —

Given,

• Quadratic equation → 2x²+px-15=0
• It's root → -5

So,

2 × (-5) + p×(-5) -15 = 0

→ -10 -5p -15 = 0

→ -25 - 5p = 0

→ 5p = -25

$\bold{→ \boxed {\color{red}{\bold {p=-5}}}}$

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( 2 ) —

Given,

• Quadratic equation → px² + x + k = 0

{ In the above solution we got -5 as the value of p}

So, the quadratic equation is → -5x² + x + k = 0

Again , we are given that this quadratic equation has two equal roots .

So, Discriminant must be 0 to satisfy this condition .

Now,

$\color{green} \bold{Discriminant = b^{2} - 4ac}$

$\color{blue}\bold{→ Discriminant = 1^{2} - 4 × (-5) × k }$

$\color{blue} \bold{→ Discriminant =1 - 20k }$

Again , Discriminant must be 0

$\color{indigo} \bold{→ Discriminant =0 }$

$\color{indigo} \bold{→1 - 20k =0 }$

$\color{indigo} \bold{→ 20k =1 }$

$\color{green} \bold{→ \boxed{\bold{k = \frac{1}{20} }}}$