Question

If -4 is a root of the quadratic equation \[{x^2} + px - 4 = 0\], and the quadratic equation \[{x^2} + px + k = 0\] has equal root, find the value of \[k\].

Answer 1

Hint: First of all, find the unknown value in the quadratic equation \[{x^2} + px - 4 = 0\]. Then substitute this value in the quadratic equation \[{x^2} + px + k = 0\] to obtain the expression in one variable. Then equate its discriminant to zero as they have equal roots to find the value of \[k\].

Complete Step-by-Step solution:
Given -4 is a root of the quadratic equation \[{x^2} + px - 4 = 0\].
Let the other root of the quadratic equation \[{x^2} + px - 4 = 0\] be \[A\].
We know that for the quadratic equation \[a{x^2} + bx + c = 0\], the sum of the roots is given by \[\dfrac{{ - b}}{a}\] and the product of the roots \[\dfrac{c}{a}\].
So, the product of the roots of the equation \[{x^2} + px - 4 = 0\] is given by
\[
  A \times \left( { - 4} \right) = \dfrac{{ - 4}}{1} \\
   - 4A = - 4 \\
  \therefore A = 1 \\
\]
So, the roots of the quadratic equation \[{x^2} + px - 4 = 0\] are \[ - 4,1\]
Now the sum of the roots of the quadratic equation \[{x^2} + px - 4 = 0\] is given by
\[
   - 4 + 1 = \dfrac{{ - p}}{1} \\
   - 3 = - p \\
  \therefore p = 3 \\
\]
Substituting the value \[p = 3\] in the quadratic equation \[{x^2} + px + k = 0\] the equation will change as \[{x^2} + 3x + k = 0\].
Given that for this quadratic equation \[{x^2} + 3x + k = 0\] the roots are equal.
We know that for the quadratic equation \[a{x^2} + bx + c = 0\], if the roots are equal then the discriminant of the equation \[a{x^2} + bx + c = 0\] is equal to zero i.e., \[D = {b^2} - 4ac = 0\].
So, the discriminant of the quadratic equation \[{x^2} + 3x + k = 0\] is equal to zero.
\[
  D = {\left( 3 \right)^2} - 4\left( 1 \right)\left( k \right) = 0 \\
  9 - 4k = 0 \\
  4k = 9 \\
  \therefore k = \dfrac{9}{4} \\
\]
Thus, the value of \[k\] is \[\dfrac{9}{4}\].

Note: The quadratic equation \[a{x^2} + bx + c = 0\], the sum of the roots is given by \[\dfrac{{ - b}}{a}\] , the product of the roots \[\dfrac{c}{a}\] and if this quadratic equation have equal roots then the discriminant of the equation \[a{x^2} + bx + c = 0\] is equal to zero i.e., \[D = {b^2} - 4ac = 0\].