Question

How do you use the discriminant to find all values of b for which the equation \[2{{x}^{2}}-bx-9=0\] has one real root?

Answer 1

Hint: In this problem, we have to find the value of b, using the discriminant formula. We are given that the equation has one solution or real root, the discriminant will be equal to 0. We can then substitute the values for a, c from the given equation in the discriminant formula to find the value of b.

Complete step-by-step answer:
We know that the given equation is,
\[2{{x}^{2}}-bx-9=0\]
We know that the discriminant formula is,
\[\Delta ={{b}^{2}}-4ac\]
We can see that the given equation has one root, then the discriminant value will be 0.
\[\Delta =0\]
We can now write it as,
\[\Rightarrow {{b}^{2}}-4ac=0\]….. (1)
We know that the given equation is \[2{{x}^{2}}-bx-9=0\],
Where a = 2, b = -b, c = -9.
We can now substitute the values of a, b, c in the formula (1), we get
\[\Rightarrow -{{b}^{2}}-4\left( 2 \right)\left( -9 \right)=0\]
We can now simplify the above step, we get
\[\Rightarrow -{{b}^{2}}=-72\]
We can now cancel the negative sign on both sides and we can take square root on both sides, we get
\[\begin{align}
  & \Rightarrow b=\pm \sqrt{72} \\
 & \Rightarrow b=\pm 6\sqrt{2} \\
\end{align}\]
Therefore, the value of \[b=\pm 6\sqrt{2}\].

Note: We should always remember that if the discriminant value is equal to 0 then we have one solution, if the discriminant is positive then we will have two real roots, if the discriminant is negative, then we will have complex numbers as the solution.