Question

How do you solve $4{x^2} + 8x + 4 = 0?$

Answer 1

Hint: This is a two degree one variable equation, or simply a quadratic equation. It can be solved with the help of a quadratic formula for that first find the discriminant of the given quadratic equation and then put the values directly in the roots formula of quadratic equation to get the required solution. You will get two solutions for a quadratic equation.

Formula Used:
For a quadratic equation $a{x^2} + bx + c = 0$, discriminant is given as follows
$D = {b^2} - 4ac$
And its roots are given as
$x = \dfrac{{ - b \pm \sqrt D }}{{2a}}$
Use this information to solve the question.$x = \dfrac{{ - b \pm \sqrt D }}{{2a}}$

Complete step by step solution:
To solve the equation $4{x^2} + 8x + 4 = 0$, we will use quadratic formula and for that we will first find the discriminant of the given equation
Discriminant of a quadratic equation $a{x^2} + bx + c = 0$, is calculated as follows
$D = {b^2} - 4ac$
And roots are given as
$x = \dfrac{{ - b \pm \sqrt D }}{{2a}}$
For the equation $4{x^2} + 8x + 4 = 0$ values of $a,\;b\;{\text{and}}\;c$ are equals to $4,\;8\;{\text{and}}\;4$ respectively,
$\therefore $ The discriminant for given equation will be
$
 \Rightarrow D = {8^2} - 4 \times 4 \times 4 \\
  \Rightarrow 64 - 64 \\
  \Rightarrow 0 \\
 $
Now finding the solution, by putting all respective values in the root formula
$
\Rightarrow x = \dfrac{{ - b \pm \sqrt D }}{{2a}} \\
\Rightarrow \dfrac{{ - 8 \pm \sqrt 0 }}{{2 \times 4}} \\
 \Rightarrow \dfrac{{ - 8 \pm 0}}{2} \\
 \Rightarrow \dfrac{{ - 8}}{2} \\
  \Rightarrow - 4 \\
 $
So the required root for the given equation is $x = - 4$

Note: We get only one root for the solution of given quadratic equation because the discriminant of the given quadratic equation is equal to zero which means that the roots of the quadratic equation will be real and unique (that is both roots will be the same). We can understand this with the graph of equation of too graph of the given quadratic equation will touch the axis only once which is at the point $x = - 4$ that says its root is $x = - 4$