Question

Find the value of $p$ if $2{x^2} + px + 8 = 0$ has equal roots?

Answer 1

Any equation in the form of $a{x^2} + bx + c = 0$ is said to be in quadratic equation. Where $x$is a variable and a, b, and c are known variables. Where$a \ne 0$, if $a = 0$ the equation will be a linear equation. Quadratic equations can be solved by the quadratic formula known as the complete square method or sridharacharya formula.

Complete step-by-step solution:
Given,
$2{x^2} + px + 8 = 0$
Here
$a = 2$
$b = p$
$c = 8$
As we know that
$\therefore D = {b^2} - 4ac$
This is called discriminant which is responsible for the nature of roots of a quadratic equation, as
$D = {b^2} - 4ac >0$ , then roots will be real and different.
$D = {b^2} - 4ac = 0$ , then roots will be real and equal.
$D = {b^2} - 4ac < 0$ , then roots will be imaginary and conjugates.
From the above condition $D = 0$ we get the equal roots, So
$\Rightarrow {b^2} - 4ac = 0$
Put the values
$\Rightarrow {(p)^2} - 4 \times 2 \times 8 = 0$
$\Rightarrow {(p)^2} = 8 \times 8$
$\Rightarrow {(p)^2} = {(8)^2}$s
$\Rightarrow p = \pm 8$
Hence the value of $p = \pm 8$.

Note: The highest point or lowest point of parabola is found by using quadratic equation, which is known as the vertex. For example hitting the golf ball. A formula that yields the zeros of any parabola, the quadratic formula can also be used to identify the axis of symmetry of the parabola. And the number of real zeros of the quadratic equation.