Question Solve the quadratic equation $9x^2 + 60x + 95 = -5$ for real solutions.

Studdy Solution

STEP 1

Assumptions
1. The equation given is a quadratic equation in the form of $ax^{2} + bx + c = 0$
2. The coefficients are real numbers
3. We are looking for real solutions

STEP 2

First, we need to rewrite the equation in the standard form of a quadratic equation $ax^{2} + bx + c = 0$ . We can do this by adding 5 to both sides of the equation.
$9x^{2} + 60x + 95 + 5 = 0$

STEP 3

Simplify the equation.
$9x^{2} + 60x + 100 = 0$

STEP 4

Now, we can solve the quadratic equation using the quadratic formula $x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$

STEP 5

Plug in the values for a, b, and c into the quadratic formula.
$x = \frac{-60 \pm \sqrt{(60)^{2} - 49100}}{2*9}$

STEP 6

Simplify the equation inside the square root.
$x = \frac{-60 \pm \sqrt{3600 - 3600}}{18}$

STEP 7

Calculate the value inside the square root.
$x = \frac{-60 \pm \sqrt{0}}{18}$

STEP 8

Simplify the equation.
$x = \frac{-60 \pm 0}{18}$

STEP 9

Calculate the two possible values for x.
$x = \frac{-60 + 0}{18}, \frac{-60 - 0}{18}$

STEP 10

Simplify to find the solutions for x.
$x = \frac{-60}{18}, \frac{-60}{18}$

STEP 11

Calculate the solutions for x.
$x = -\frac{60}{18}, -\frac{60}{18}$

STEP 12

Simplify the solutions for x.
$x = -\frac{20}{6}, -\frac{20}{6}$

STEP 13

Simplify the solutions for x.
$x = -\frac{10}{3}, -\frac{10}{3}$
Since both solutions are the same, we only have one solution for this quadratic equation.
The solution to the equation $9x^{2} + 60x + 95 = -5$ is $x = -\frac{10}{3}$ .

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