Studdy AI Logo
Math

Question Solve the quadratic equation 9x2+12x=−69x^2 + 12x = -6 for the value of xx.

Studdy Solution

STEP 1

Assumptions
1. We have a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0.
2. The given equation is 9x2+12x=−69x^2 + 12x = -6.
3. We will solve for xx by first bringing the equation to standard form and then using the quadratic formula or factoring, depending on the situation.

STEP 2

First, we need to bring the equation to the standard form of a quadratic equation, which is ax2+bx+c=0ax^2 + bx + c = 0. To do this, we will add 66 to both sides of the equation to move the constant term to the left side.
9x2+12x+6=09x^2 + 12x + 6 = 0

STEP 3

Now, we will attempt to factor the quadratic if possible. We look for two numbers that multiply to 9⋅6=549 \cdot 6 = 54 and add up to 1212.

STEP 4

We observe that the numbers 66 and 99 satisfy these conditions because 6⋅9=546 \cdot 9 = 54 and 6+9=156 + 9 = 15. However, since we need the numbers to add up to 1212, not 1515, we realize that factoring might not be straightforward. We will use the quadratic formula instead.

STEP 5

The quadratic formula is given by:
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
where aa, bb, and cc are the coefficients from the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0.

STEP 6

Identify the coefficients aa, bb, and cc from the standard form of the equation.
a=9,b=12,c=6a = 9, \quad b = 12, \quad c = 6

STEP 7

Substitute the coefficients aa, bb, and cc into the quadratic formula.
x=−12±122−4⋅9⋅62⋅9x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 9 \cdot 6}}{2 \cdot 9}

STEP 8

Calculate the discriminant, which is the part under the square root in the quadratic formula: b2−4acb^2 - 4ac.
Discriminant=122−4⋅9⋅6Discriminant = 12^2 - 4 \cdot 9 \cdot 6

STEP 9

Compute the value of the discriminant.
Discriminant=144−216Discriminant = 144 - 216

STEP 10

Calculate the discriminant.
Discriminant=144−216=−72Discriminant = 144 - 216 = -72

STEP 11

Since the discriminant is negative, there are no real solutions to the equation. The solutions will be complex numbers.

STEP 12

Continue with the quadratic formula using the negative discriminant to find the complex solutions.
x=−12±−7218x = \frac{-12 \pm \sqrt{-72}}{18}

STEP 13

Simplify the square root of the negative number by introducing the imaginary unit ii, where i2=−1i^2 = -1.
−72=72⋅i\sqrt{-72} = \sqrt{72} \cdot i

STEP 14

Find the square root of 7272. Since 72=36⋅272 = 36 \cdot 2 and 3636 is a perfect square, we have:
72=36⋅2=62\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}

STEP 15

Substitute the square root back into the quadratic formula.
x=−12±62i18x = \frac{-12 \pm 6\sqrt{2}i}{18}

STEP 16

Simplify the fraction by dividing both the numerator and the denominator by 66.
x=−2±2i3x = \frac{-2 \pm \sqrt{2}i}{3}

STEP 17

Write the final solutions, which are complex numbers.
x=−23+23iorx=−23−23ix = \frac{-2}{3} + \frac{\sqrt{2}}{3}i \quad \text{or} \quad x = \frac{-2}{3} - \frac{\sqrt{2}}{3}i
The solutions to the equation 9x2+12x=−69x^2 + 12x = -6 are:
x=−23+23iandx=−23−23ix = \frac{-2}{3} + \frac{\sqrt{2}}{3}i \quad \text{and} \quad x = \frac{-2}{3} - \frac{\sqrt{2}}{3}i

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord