Question

How do you solve $2{{x}^{2}}-28x+98=0$ using the quadratic formula?

Answer 1

Hint: In this question, we have to find the value of x. Thus, we will apply the discriminant formula to solve this problem. We will first compare the given equation with the general form of the quadratic equation and thus get the value of a, b, and c. Then, we will find the value of discriminant $D=\sqrt{{{b}^{2}}-4ac}$, and then apply the discriminant formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ to get an equation. After the necessary calculations, we will get two equations for x, so we solve them separately to get the value of x, which is our required answer.

Complete step by step solution:
According to the question, a quadratic equation is given to us and we have to solve the equation for the value of x.
The equation is $2{{x}^{2}}-28x+98=0$ ----------------- (1)
As we know, the general form of quadratic equation is $a{{x}^{2}}+bx+c=0$ ---------- (2)
Thus, on comparing equation (1) and (2), we get $a=2,$ $b=-28,$ and $c=98$ ------- (3)
So, now we will apply the discriminant formula $D=\sqrt{{{b}^{2}}-4ac}$ by putting the above values in the formula, we get
$\begin{align}
  & \Rightarrow D=\sqrt{{{(-28)}^{2}}-4.(2).(98)} \\
 & \Rightarrow D=\sqrt{784-784} \\
\end{align}$
Thus, on further solving, we get
$\Rightarrow D=\sqrt{0}$
$\Rightarrow D=0$ -------------- (4)
Thus, we see that the value of discriminant is equal to zero, which implies we have the same roots for the given equation.
Now, we will apply the discriminant formula , which is
$\Rightarrow x=\dfrac{-b\pm D}{2a}$
$\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ --------------- (5)
So, we will put the value of equation (3) and (4) in equation (5), we get
$\Rightarrow x=\dfrac{-(-28)\pm 0}{2.(2)}$
On further simplification, we get
$\Rightarrow x=\dfrac{28}{4}$
Therefore, on further simplification, we get
$\Rightarrow x=7$

Therefore, for the equation $2{{x}^{2}}-28x+98=0$, we get the value of $x=7,7$

Note: While solving this problem, do all the steps carefully and avoid mathematical errors to get the correct answer. Always remember when the value of discriminant is equal to 0, then the roots are real and equal to each other.