Solved on Dec 17, 2023

Solve the quadratic equation $9x^2 + 12x = -6$ for the value of $x$ .

STEP 1

Assumptions
1. We have a quadratic equation in the form $ax^2 + bx + c = 0$ .
2. The given equation is $9x^2 + 12x = -6$ .
3. We will solve for $x$ 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 $ax^2 + bx + c = 0$ . To do this, we will add $6$ to both sides of the equation to move the constant term to the left side.
$9x^2 + 12x + 6 = 0$

STEP 3

Now, we will attempt to factor the quadratic if possible. We look for two numbers that multiply to $9 \cdot 6 = 54$ and add up to $12$ .

STEP 4

We observe that the numbers $6$ and $9$ satisfy these conditions because $6 \cdot 9 = 54$ and $6 + 9 = 15$ . However, since we need the numbers to add up to $12$ , not $15$ , we realize that factoring might not be straightforward. We will use the quadratic formula instead.

STEP 5

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

STEP 6

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

STEP 7

Substitute the coefficients $a$ , $b$ , and $c$ into the quadratic formula.
$x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 9 \cdot 6}}{2 \cdot 9}$

STEP 8

Calculate the discriminant, which is the part under the square root in the quadratic formula: $b^2 - 4ac$ .
$Discriminant = 12^2 - 4 \cdot 9 \cdot 6$

STEP 9

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

STEP 10

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

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.
$x = \frac{-12 \pm \sqrt{-72}}{18}$

STEP 13

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

STEP 14

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

STEP 15

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

STEP 16

Simplify the fraction by dividing both the numerator and the denominator by $6$ .
$x = \frac{-2 \pm \sqrt{2}i}{3}$

STEP 17

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

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Solve the quadratic equation $9x^2 + 12x = -6$ for the value of $x$ .

STEP 1

Assumptions
1. We have a quadratic equation in the form $ax^2 + bx + c = 0$ .
2. The given equation is $9x^2 + 12x = -6$ .
3. We will solve for $x$ 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 $ax^2 + bx + c = 0$ . To do this, we will add $6$ to both sides of the equation to move the constant term to the left side.
$9x^2 + 12x + 6 = 0$

STEP 3

Now, we will attempt to factor the quadratic if possible. We look for two numbers that multiply to $9 \cdot 6 = 54$ and add up to $12$ .

STEP 4

We observe that the numbers $6$ and $9$ satisfy these conditions because $6 \cdot 9 = 54$ and $6 + 9 = 15$ . However, since we need the numbers to add up to $12$ , not $15$ , we realize that factoring might not be straightforward. We will use the quadratic formula instead.

STEP 5

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

STEP 6

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

STEP 7

Substitute the coefficients $a$ , $b$ , and $c$ into the quadratic formula.
$x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 9 \cdot 6}}{2 \cdot 9}$

STEP 8

Calculate the discriminant, which is the part under the square root in the quadratic formula: $b^2 - 4ac$ .
$Discriminant = 12^2 - 4 \cdot 9 \cdot 6$

STEP 9

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

STEP 10

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

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.
$x = \frac{-12 \pm \sqrt{-72}}{18}$

STEP 13

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

STEP 14

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

STEP 15

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

STEP 16

Simplify the fraction by dividing both the numerator and the denominator by $6$ .
$x = \frac{-2 \pm \sqrt{2}i}{3}$

STEP 17

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

STEP 1

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 = 0ax2+bx+c=0. To do this, we will add 666 to both sides of the equation to move the constant term to the left side.9x2+12x+6=09x^2 + 12x + 6 = 09x2+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 = 549⋅6=54 and add up to 121212.

STEP 4

We observe that the numbers 666 and 999 satisfy these conditions because 6⋅9=546 \cdot 9 = 546⋅9=54 and 6+9=156 + 9 = 156+9=15. However, since we need the numbers to add up to 121212, not 151515, 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}x=2a−b±b2−4ac​​where aaa, bbb, and ccc are the coefficients from the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.

STEP 6

Identify the coefficients aaa, bbb, and ccc from the standard form of the equation.a=9,b=12,c=6a = 9, \quad b = 12, \quad c = 6a=9,b=12,c=6

STEP 7

Substitute the coefficients aaa, bbb, and ccc 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}x=2⋅9−12±122−4⋅9⋅6​​

STEP 8

Calculate the discriminant, which is the part under the square root in the quadratic formula: b2−4acb^2 - 4acb2−4ac.Discriminant=122−4⋅9⋅6Discriminant = 12^2 - 4 \cdot 9 \cdot 6Discriminant=122−4⋅9⋅6

STEP 9

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

STEP 10

Calculate the discriminant.Discriminant=144−216=−72Discriminant = 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}x=18−12±−72​​

STEP 13

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

STEP 14

Find the square root of 727272. Since 72=36⋅272 = 36 \cdot 272=36⋅2 and 363636 is a perfect square, we have:72=36⋅2=62\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}72​=36⋅2​=62​

STEP 15

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

STEP 16

Simplify the fraction by dividing both the numerator and the denominator by 666.x=−2±2i3x = \frac{-2 \pm \sqrt{2}i}{3}x=3−2±2​i​

STEP 17

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Solve the quadratic equation $9x^2 + 12x = -6$ for the value of $x$ .

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

Now, we will attempt to factor the quadratic if possible. We look for two numbers that multiply to $9 \cdot 6 = 54$ and add up to $12$ .

We observe that the numbers $6$ and $9$ satisfy these conditions because $6 \cdot 9 = 54$ and $6 + 9 = 15$ . However, since we need the numbers to add up to $12$ , not $15$ , we realize that factoring might not be straightforward. We will use the quadratic formula instead.

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

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

Substitute the coefficients $a$ , $b$ , and $c$ into the quadratic formula.
$x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 9 \cdot 6}}{2 \cdot 9}$

Calculate the discriminant, which is the part under the square root in the quadratic formula: $b^2 - 4ac$ .
$Discriminant = 12^2 - 4 \cdot 9 \cdot 6$

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

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

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

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

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

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

Simplify the fraction by dividing both the numerator and the denominator by $6$ .
$x = \frac{-2 \pm \sqrt{2}i}{3}$