Question

How do you solve $-4{{x}^{2}}+x+1=0$ using the quadratic equation?

Answer 1

Hint: In this question, we have to find the value of x. The equation given to us is in the form of a quadratic, therefore when we solve this problem, we will get two values for x, which satisfy the equation. Therefore, we will apply the discriminant method to solve this problem. We will compare the general form of quadratic equation and the given equation to get the value of a, b, and c. Then, we will get the value of discriminant $D=\sqrt{{{b}^{2}}-4ac}$, and thus find the value of x using the discriminant formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .After necessary calculations, get two equations of x, we solve them separately to get the value of x, which is our required answer.

Complete step-by-step answer:
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 $-4{{x}^{2}}+x+1=0$ ----------------- (1)
As we know, the general quadratic equation is in form of $a{{x}^{2}}+bx+c=0$ ---------- (2)
Thus, on comparing equation (2) and (3), we get $a=-4,$ $b=1,$ and $c=1$ ------- (3)
So, now we will apply the discriminant formula $D=\sqrt{{{b}^{2}}-4ac}$ by putting the above values in the formula, we get
$\begin{align}
  & \Rightarrow D=\sqrt{{{(1)}^{2}}-4.(-4).(1)} \\
 & \Rightarrow D=\sqrt{1-(-16)} \\
\end{align}$
Thus, on further solving, we get
$\Rightarrow D=\sqrt{1+16}$
$\Rightarrow D=\sqrt{17}$ -------------- (4)
Since we see the discriminant is a real number, thus now we will find the value of x, using the formula,
$\Rightarrow x=\dfrac{-b\pm D}{2a}$
$\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ --------------- (5)
So, we will put the value of equation (3) and (4) in equation (5), we get
$\Rightarrow x=\dfrac{-(1)\pm \sqrt{17}}{2.(-4)}$
On further simplification, we get
$\Rightarrow x=\dfrac{-1\pm \sqrt{17}}{-8}$
Therefore, we will split the above equation in terms of (+) and (-), we get
$\Rightarrow x=\dfrac{-1+\sqrt{17}}{-8}$ -------- (6) , or
 $\Rightarrow x=\dfrac{-1-\sqrt{17}}{-8}$ ---------- (7)
Now, we will first solve equation (6), we get
$\Rightarrow x=\dfrac{1}{8}-\dfrac{\sqrt{17}}{8}$
Now, we will first solve equation (7), we get
$\Rightarrow x=\dfrac{1}{8}+\dfrac{\sqrt{17}}{8}$

Therefore, for the equation $-4{{x}^{2}}+x+1=0$ , we get the value of $x=\dfrac{1}{8}-\dfrac{\sqrt{17}}{8},\dfrac{1}{8}+\dfrac{\sqrt{17}}{8}$ .

Note: While solving this problem, do all the steps carefully and avoid errors to get the correct answer. Do mention the formula you are using. Always first find the value of discriminant and then apply it in the formula to get an accurate answer.