Question

How do you solve $4{x^2} - 64 = 0$ using the quadratic formula?

Answer 1

Hint:Compare the given equation to the general form of a quadratic equation and then write the values of terms accordingly. Put the values in general solution of the quadratic equation, finally you will get the solution for your equation
General form a quadratic equation is given as following:
$a{x^2} + bx + c = 0,\;{\text{where}}\;a,\;b\;{\text{and}}\;c \in I$
Its solution will be given as $x = \dfrac{{ - b \pm \sqrt D }}{{2a}},$ where “D” is the discriminant of the quadratic equation which could be find as $D = {b^2} - 4ac$

Complete step by step answer:
To solve $4{x^2} - 64 = 0$ with the help of quadratic formula, let us first compare the quadratic coefficients to the given equation
$a{x^2} + bx + c = 0\;{\text{and}}\;4{x^2} - 64 = 0$
We can write $4{x^2} - 64 = 0\;as\;4{x^2} + 0x + ( - 64) = 0$ it will not affect the original equation.Now,
$a{x^2} + bx + c = 0\;{\text{and}}\;4{x^2} + 0x + ( - 64) = 0$
We can see from the above comparison that value of $a = 4,\;b = 0\;{\text{and}}\;c = - 64$
The general solution for a quadratic equation $a{x^2} + bx + c = 0,\;{\text{where}}\;a,\;b\;{\text{and}}\;c \in I$ is given as $x = \dfrac{{ - b \pm \sqrt D }}{{2a}},$ where “D” is the discriminant of the quadratic equation which can be calculated as $D = {b^2} - 4ac$
So in order to solve the given equation through quadratic formula, now we have to calculate the value of determinant,
$
\Rightarrow D = {b^2} - 4ac \\
\Rightarrow D = {0^2} - 4 \times 4 \times ( - 64) \\
\Rightarrow D = 1024 \\ $
Now putting the respective values in the general solution to find the solution for the given equation,
$
\Rightarrow x = \dfrac{{ - b \pm \sqrt D }}{{2a}} \\
\Rightarrow x = \dfrac{{ - 0 \pm \sqrt {1024} }}{{2 \times 4}} \\
\Rightarrow x = \dfrac{{ \pm 32}}{8} \\
\therefore x = \pm 4 \\ $
Therefore $x = \pm 4$ is the solution for the equation $4{x^2} - 64 = 0$.

Note:Discriminant of a quadratic equation gives information about the nature of its roots. If value of “D” is less than zero then the roots will be imaginary, if “D” is equals to zero then the roots will be real and equal, and if “D” is greater than zero then roots will be real and distinct.You can also solve this equation directly with the help of algebraic operations.Try it by yourself and check the answer.