Question $\frac{-7 \pm \sqrt{7)^{2}-4(-2)(-3)}}{2(-2)}$

Studdy Solution

STEP 1

1. The expression given is a quadratic formula in the form $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
2. The values of $a$ , $b$ , and $c$ are extracted from the context of the formula.
3. The expression under the square root, $b^2 - 4ac$ , is called the discriminant.

STEP 2

1. Identify the coefficients $a$ , $b$ , and $c$ .
2. Calculate the discriminant.
3. Evaluate the square root of the discriminant.
4. Substitute the values into the quadratic formula.
5. Simplify the expression to find the two possible solutions.

STEP 3

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic formula expression:
Given expression: $\frac{-7 \pm \sqrt{7^2 - 4(-2)(-3)}}{2(-2)}$
- $b = 7$ - $a = -2$ - $c = -3$

STEP 4

Calculate the discriminant, $b^2 - 4ac$ :
$b^2 - 4ac = 7^2 - 4(-2)(-3)$
$= 49 - 4 \times (-2) \times (-3)$
$= 49 - 24$
$= 25$

STEP 5

Evaluate the square root of the discriminant:
$\sqrt{25} = 5$

STEP 6

Substitute the values into the quadratic formula:
$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-7 \pm 5}{2(-2)}$

STEP 7

Simplify the expression to find the two possible solutions:
First solution:
$\frac{-7 + 5}{-4} = \frac{-2}{-4} = \frac{1}{2}$
Second solution:
$\frac{-7 - 5}{-4} = \frac{-12}{-4} = 3$
The two possible solutions are:
$\boxed{\frac{1}{2}} \quad \text{and} \quad \boxed{3}$

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