Root Formula: Solve x² + 7¾x - 2 Using Step-by-Step Method

The quadratic equation given is:

$x^2 + 7\frac{3}{4}x - 2 = 0$

First, let's rewrite the coefficient $b$ as a decimal for simplicity:

$b = 7\frac{3}{4} = 7.75$

Given $a = 1$, $b = 7.75$, and $c = -2$, we apply the quadratic formula:

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

Calculate the discriminant:

$\text{Discriminant} = b^2 - 4ac = (7.75)^2 - 4 \times 1 \times -2$

$= 60.0625 + 8 = 68.0625$

Now, find the roots by plugging into the quadratic formula:

$x = \frac{-7.75 \pm \sqrt{68.0625}}{2}$

Calculate the square root:

$\sqrt{68.0625} = 8.25$

Find the roots:

$x_1 = \frac{-7.75 + 8.25}{2} = \frac{0.5}{2} = 0.25$

$x_2 = \frac{-7.75 - 8.25}{2} = \frac{-16}{2} = -8$

Thus, the factored form of the trinomial can be written using these roots:

$(x - x_1)(x - x_2) = (x - 0.25)(x + 8)$

Rewriting with fractional representations:

$(x - \frac{1}{4})(x + 8)$

Thus, the trinomial can be expressed as:

$(x + 8)(x - \frac{1}{4})$

Comparing with the options provided, the correct answer is:

$\boxed{(x + 8)(x - \frac{1}{4})}$