Question

How do you solve $4{p^2} - 3p + 7 = 0$?

Answer 1

Hint: The given equation is a quadratic equation in one variable $p$. It should first be changed into the general form of a quadratic equation, which is given as $a{p^2} + bp + c = 0$. Solving this equation gives two values of the variable $p$ as the result. We will then use the coefficients to find the value of discriminant $D = {b^2} - 4ac$. If $D \geqslant 0$, we use the quadratic formula $p = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$to solve for $p$.

Complete step-by-step solution:
We have to solve the given equation $4{p^2} - 3p + 7 = 0$.
To find the value of $p$, we have to put the values of $a$, $b$ and $c$ in the quadratic formula. To get the values of $a$, $b$ and $c$ from the given equation, we compare it with the general form of the quadratic equation given by $a{p^2} + bp + c = 0$
On comparing the above rearranged equation with the general form, we observe that
Coefficient $a$ of ${p^2}$ is $4$,
Coefficient $b$ of $p$ is $ - 3$,
and the constant term $c$ is $7$.
Thus, $a = 4$, $b = - 3$ and $c = 7$.
Now we check the value of discriminant $D$ by using the above values. $D$ is given by,
$D = {b^2} - 4ac$
If $D > 0$, the two roots will be real and distinct.
If $D = 0$, the two roots will be real and equal.
If $D < 0$, the two roots will be imaginary and distinct.
For $a = 4$, $b = - 3$ and $c = 7$, we have:
$
  D = {b^2} - 4ac \\
   \Rightarrow D = {( - 3)^2} - 4 \times 4 \times 7 \\
   \Rightarrow D = 9 - 112 \\
   \Rightarrow D = - 103 \\
$
Since, $D$ is negative here, the roots will be imaginary in nature. Therefore, there is no real solution for the given equation.

Note: When solving a quadratic equation, discriminant of the equation should always be calculated to check if the roots are real or not. In case, $D < 0$, the real solution of the equation would not be possible. We can see that if $D < 0$, we get a negative term under the roots in the quadratic formula the real solution of which is not possible.