Solve the Quadratic Equation: x²/2 + x - 4 = 0 Step-by-Step

To solve the equation $\frac{x^2}{2} + x - 4 = 0$, we'll use the quadratic formula. First, we need to rewrite the equation in standard quadratic form, $ax^2 + bx + c = 0$.

Start by multiplying the entire equation by 2 to eliminate the fraction:

$\ x^2 + 2x - 8 = 0$

Now, identify the coefficients:

• $a = 1$
• $b = 2$
• $c = -8$

Next, plug these coefficients into the quadratic formula:

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

Substitute $a = 1$, $b = 2$, and $c = -8$ into the formula:

$\ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1}$

Simplify inside the square root first:

$\ 2^2 - 4 \cdot 1 \cdot (-8) = 4 + 32 = 36$

Substitute back:

$\ x = \frac{-2 \pm \sqrt{36}}{2}$

The square root of 36 is 6, so:

$\ x = \frac{-2 \pm 6}{2}$

Calculate the two possible solutions:

1. $x_1 = \frac{-2 + 6}{2} = \frac{4}{2} = 2$ 2. $x_2 = \frac{-2 - 6}{2} = \frac{-8}{2} = -4$

Therefore, the solutions for the equation $\frac{x^2}{2} + x - 4 = 0$ are $x_1 = 2$ and $x_2 = -4$.