The problem is to solve the quadratic equation $16x^2 + 7 = 13$ using the quadratic formula.

1. Problem Description

The problem is to solve the quadratic equation $16x^2 + 7 = 13$ using the quadratic formula.

2. Solution Steps

First, rewrite the equation in the standard quadratic form $ax^2 + bx + c = 0$.

$16x^2 + 7 = 13$

$16x^2 + 7 - 13 = 0$

$16x^2 - 6 = 0$

Thus, $a = 16$, $b = 0$, and $c = -6$.

The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substitute the values of $a$, $b$, and $c$ into the quadratic formula:

$x = \frac{-0 \pm \sqrt{0^2 - 4(16)(-6)}}{2(16)}$

$x = \frac{\pm \sqrt{0 + 384}}{32}$

$x = \frac{\pm \sqrt{384}}{32}$

Simplify the square root:

$384 = 64 \times 6$

$\sqrt{384} = \sqrt{64 \times 6} = \sqrt{64} \times \sqrt{6} = 8\sqrt{6}$

Substitute this back into the expression for $x$:

$x = \frac{\pm 8\sqrt{6}}{32}$

$x = \frac{\pm \sqrt{6}}{4}$

3. Final Answer

$x = \frac{\sqrt{6}}{4}, -\frac{\sqrt{6}}{4}$

$x = \pm \frac{\sqrt{6}}{4}$