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Question: Find the roots of the quadratic equation \(16{{x}^{2}}-24x-1=0\) by using the quadratic formula....

Find the roots of the quadratic equation 16x2−24x−1=016{{x}^{2}}-24x-1=0 by using the quadratic formula.

Explanation

Solution

We will be using the quadratic formula to find the roots of the given quadratic equation. The general quadratic equation is of the form ax2+bx+c=0a{{x}^{2}}+bx+c=0, where a≠0a\ne 0. We know that we can obtain the roots of a quadratic equation by using the quadratic formula. The quadratic formula is given by
x=−b±b2−4ac2ax=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}.

Complete step-by-step solution:
The given quadratic equation is 16x2−24x−1=016{{x}^{2}}-24x-1=0. The general quadratic equation is ax2+bx+c=0a{{x}^{2}}+bx+c=0. Comparing the given quadratic equation with the general quadratic equation, we will get the values for aa, bb and cc. We have the following values for aa, bb and cc:

& a=16 \\\ & b=-24 \\\ & c=-1 \\\ \end{aligned}$$ The quadratic formula to obtain the roots of a general quadratic equation is given by $$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$$ We have the formula and the values for $a$, $b$ and $c$. Now, we will substitute the values for $a$ , $b$ and $c$ in this formula. So we have the following equation, $$x=\dfrac{-(-24)\pm \sqrt{{{(-24)}^{2}}-4\cdot (16)\cdot (-1)}}{2\cdot 16}$$ We will simplify the above equation to obtain the roots of the quadratic equation in the following manner, $$x=\dfrac{24\pm \sqrt{576+64}}{32}$$ $$\begin{aligned} & x=\dfrac{24\pm \sqrt{640}}{32} \\\ & =\dfrac{24\pm 8\sqrt{10}}{32} \end{aligned}$$ We can simplify the above equation by factoring the numerator. We get the following expression, $$\begin{aligned} & x=\dfrac{8\cdot (3\pm \sqrt{10})}{32} \\\ & =\dfrac{3\pm \sqrt{10}}{4} \end{aligned}$$ So we have obtained the two roots that have values $\dfrac{3+\sqrt{10}}{4}$ and $\dfrac{3-\sqrt{10}}{4}$ . **Note:** The signs of the values of $a$ ,$b$ and $c$ must be carefully written. It is possible to misplace the signs while substituting the values in the quadratic formula. We can find the roots of a quadratic equation by other methods, such as factorization method, completing square method or drawing a graph.