Question

MULTIPLE CHOICE Which of the following is a solution of the equation \(2 x^{2}+8 x-25=5 ?\) \begin{array}{ccccc}\mathbf{A} & \sqrt{17}+1 & \mathbf{B} & -\sqrt{19}-2 & \mathbf{C} & \sqrt{17}-2 & \mathbf{D} &\sqrt{21} & -2\end{array}

Short answer

The correct answer is A: \(-1 + \sqrt{17}\)

Step-by-step solution

Write out the equation

The original equation is \(2x^{2}+8x-25=5\). Let's first reformat this into a standard quadratic equation \(ax^{2}+bx+c = 0\) by moving 5 to the left side of the equation.

Reformulate the equation

So our new equation becomes \(2x^{2}+8x-30 = 0\). Now we can identify \(a=2\), \(b=8\), and \(c=-30\).

Apply the quadratic formula

Let's solve for \(x\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting the values we found for \(a\), \(b\), and \(c\) into the formula gives us: \(x = \frac{-8 \pm \sqrt{(8)^2 - 4*2*(-30)}}{2*2}\).

Evaluate the expression

Solving the above formula gives: \(x = -1 \pm \sqrt{17}\). This means the solutions to the equation are \(x_1 = -1 + \sqrt{17}\) and \(x_2 = -1 - \sqrt{17}\).

Find the solution in the options

Looking at the answer choices, we see that both of our solutions are present. However, in the original question they've asked for a single solution, not two. So, if we look at the answer options, we can see \(x_1=-1 + \sqrt{17}\) matches option A.