Question Solve the quadratic equation $\frac{1}{2} x^{2}-3 x-\frac{7}{2}=0$ for $x$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the quadratic equation $\frac{1}{2} x^{2}-3 x-\frac{7}{2}=0$ .
2. We need to solve for $x$ .

STEP 2

First, we can multiply the entire equation by 2 to get rid of the fractions. This will make the equation easier to work with.
$2 \times \left(\frac{1}{2} x^{2}-3 x-\frac{7}{2}\right) = 2 \times 0$

STEP 3

Distribute the 2 across the terms in the parentheses.
$2 \times \frac{1}{2} x^{2} - 2 \times 3 x - 2 \times \frac{7}{2} = 0$

STEP 4

Simplify each term.
$x^{2} - 6x - 7 = 0$

STEP 5

Now we have a standard quadratic equation in the form $ax^{2} + bx + c = 0$ . We can solve for $x$ using the quadratic formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

STEP 6

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation $x^{2} - 6x - 7 = 0$ .
$a = 1, b = -6, c = -7$

STEP 7

Plug the coefficients into the quadratic formula.
$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-7)}}{2(1)}$

STEP 8

Simplify the expression inside the square root and the constants outside the square root.
$x = \frac{6 \pm \sqrt{36 + 28}}{2}$

STEP 9

Continue simplifying the expression under the square root.
$x = \frac{6 \pm \sqrt{64}}{2}$

STEP 10

Calculate the square root of 64.
$x = \frac{6 \pm 8}{2}$

STEP 11

Now we have two possible solutions for $x$ since we have a plus-minus sign.
$x = \frac{6 + 8}{2} \quad \text{or} \quad x = \frac{6 - 8}{2}$

STEP 12

Calculate the two possible values for $x$ .
$x = \frac{14}{2} \quad \text{or} \quad x = \frac{-2}{2}$

STEP 13

Simplify both fractions to find the two solutions for $x$ .
$x = 7 \quad \text{or} \quad x = -1$
The solutions to the equation $\frac{1}{2} x^{2}-3 x-\frac{7}{2}=0$ are $x = 7$ and $x = -1$ .

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