Question

Solve the equation using the Quadratic Formula. 2x(4x+1)=11
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
1)x=____ (simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.)
2)There are no real solutions.

Answer 1

To solve the equation \(2x(4x+1)=11\) using the quadratic formula, we first need to rearrange the equation into standard form.

1. Expand the left side:
\[
8x^2 + 2x = 11
\]

2. Rearrange it to set it to 0:
\[
8x^2 + 2x - 11 = 0
\]

Now we identify the coefficients \(a\), \(b\), and \(c\) in the standard form \(ax^2 + bx + c = 0\):
- \(a = 8\)
- \(b = 2\)
- \(c = -11\)

Next, we apply the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting the values of \(a\), \(b\), and \(c\) into the formula:
1. Calculate the discriminant:
\[
b^2 - 4ac = 2^2 - 4 \cdot 8 \cdot (-11) = 4 + 352 = 356
\]

2. Now plug the values into the quadratic formula:
\[
x = \frac{-2 \pm \sqrt{356}}{2 \cdot 8}
\]

3. Simplify:
\[
x = \frac{-2 \pm \sqrt{356}}{16}
\]

Next, simplify \(\sqrt{356}\). Since \(356 = 4 \cdot 89\), we have:
\[
\sqrt{356} = \sqrt{4} \cdot \sqrt{89} = 2\sqrt{89}
\]

So we substitute back:
\[
x = \frac{-2 \pm 2\sqrt{89}}{16}
\]

Factor out the 2 in the numerator:
\[
x = \frac{2(-1 \pm \sqrt{89})}{16}
\]
\[
x = \frac{-1 \pm \sqrt{89}}{8}
\]

Thus, the solutions are:
\[
x = \frac{-1 + \sqrt{89}}{8}, \quad x = \frac{-1 - \sqrt{89}}{8}
\]

Finally, the full answer is:

1) \(x = \frac{-1 + \sqrt{89}}{8}, \frac{-1 - \sqrt{89}}{8}\)