Math / Algebra

QuestionChasity is asked to solve the equation $\frac{81^{x(x+2)}}{3^{5 x-7}}=9^{x-8}$ . She is able to simplify the equation to the form $4 x^{2}+b x+c=0$ . the value of $c$ is $-15$ 1 23 9

Studdy Solution

STEP 1

1. We are given an equation involving exponents and need to simplify it to a quadratic form.
2. The bases of the exponents can be rewritten as powers of 3.
3. The simplified quadratic equation is in the form $4x^2 + bx + c = 0$ .

STEP 2

1. Rewrite all terms with a common base.
2. Simplify the equation using properties of exponents.
3. Rearrange the equation into a quadratic form.
4. Identify the value of $c$ .

STEP 3

Rewrite each term with a base of 3:
- $81 = 3^4$ , so $81^{x(x+2)} = (3^4)^{x(x+2)} = 3^{4x(x+2)}$ . - $9 = 3^2$ , so $9^{x-8} = (3^2)^{x-8} = 3^{2(x-8)}$ .
The equation becomes:
$\frac{3^{4x(x+2)}}{3^{5x-7}} = 3^{2(x-8)}$

STEP 4

Apply the properties of exponents to simplify:
$3^{4x(x+2) - (5x-7)} = 3^{2(x-8)}$
This simplifies to:
$4x(x+2) - 5x + 7 = 2(x-8)$

STEP 5

Expand and simplify both sides:
Left side:
$4x(x+2) - 5x + 7 = 4x^2 + 8x - 5x + 7 = 4x^2 + 3x + 7$
Right side:
$2(x-8) = 2x - 16$
Equating both sides:
$4x^2 + 3x + 7 = 2x - 16$
Rearrange to form a quadratic equation:
$4x^2 + 3x + 7 - 2x + 16 = 0$
Simplify:
$4x^2 + x + 23 = 0$

STEP 6

Identify the value of $c$ in the quadratic equation $4x^2 + x + 23 = 0$ .
The value of $c$ is:
$\boxed{23}$

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QuestionChasity is asked to solve the equation $\frac{81^{x(x+2)}}{3^{5 x-7}}=9^{x-8}$ . She is able to simplify the equation to the form $4 x^{2}+b x+c=0$ . the value of $c$ is $-15$ 1 23 9

Studdy Solution

STEP 1

1. We are given an equation involving exponents and need to simplify it to a quadratic form.
2. The bases of the exponents can be rewritten as powers of 3.
3. The simplified quadratic equation is in the form $4x^2 + bx + c = 0$ .

STEP 2

1. Rewrite all terms with a common base.
2. Simplify the equation using properties of exponents.
3. Rearrange the equation into a quadratic form.
4. Identify the value of $c$ .

STEP 3

Rewrite each term with a base of 3:
- $81 = 3^4$ , so $81^{x(x+2)} = (3^4)^{x(x+2)} = 3^{4x(x+2)}$ . - $9 = 3^2$ , so $9^{x-8} = (3^2)^{x-8} = 3^{2(x-8)}$ .
The equation becomes:
$\frac{3^{4x(x+2)}}{3^{5x-7}} = 3^{2(x-8)}$

STEP 4

Apply the properties of exponents to simplify:
$3^{4x(x+2) - (5x-7)} = 3^{2(x-8)}$
This simplifies to:
$4x(x+2) - 5x + 7 = 2(x-8)$

STEP 5

Expand and simplify both sides:
Left side:
$4x(x+2) - 5x + 7 = 4x^2 + 8x - 5x + 7 = 4x^2 + 3x + 7$
Right side:
$2(x-8) = 2x - 16$
Equating both sides:
$4x^2 + 3x + 7 = 2x - 16$
Rearrange to form a quadratic equation:
$4x^2 + 3x + 7 - 2x + 16 = 0$
Simplify:
$4x^2 + x + 23 = 0$

STEP 6

Identify the value of $c$ in the quadratic equation $4x^2 + x + 23 = 0$ .
The value of $c$ is:
$\boxed{23}$

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QuestionChasity is asked to solve the equation 81x(x+2)35x−7=9x−8\frac{81^{x(x+2)}}{3^{5 x-7}}=9^{x-8}35x−781x(x+2)​=9x−8. She is able to simplify the equation to the form 4x2+bx+c=04 x^{2}+b x+c=04x2+bx+c=0. the value of ccc is −15-15−15 1 23 9

Studdy Solution

STEP 1

1. We are given an equation involving exponents and need to simplify it to a quadratic form.2. The bases of the exponents can be rewritten as powers of 3.3. The simplified quadratic equation is in the form 4x2+bx+c=04x^2 + bx + c = 04x2+bx+c=0.

STEP 2

1. Rewrite all terms with a common base.2. Simplify the equation using properties of exponents.3. Rearrange the equation into a quadratic form.4. Identify the value of ccc.

STEP 3

STEP 4

Apply the properties of exponents to simplify:34x(x+2)−(5x−7)=32(x−8)3^{4x(x+2) - (5x-7)} = 3^{2(x-8)}34x(x+2)−(5x−7)=32(x−8)This simplifies to:4x(x+2)−5x+7=2(x−8)4x(x+2) - 5x + 7 = 2(x-8)4x(x+2)−5x+7=2(x−8)

STEP 5

STEP 6

Identify the value of ccc in the quadratic equation 4x2+x+23=04x^2 + x + 23 = 04x2+x+23=0.The value of ccc is:23 \boxed{23} 23​

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Studdy solves anything!

Start learning now

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

QuestionChasity is asked to solve the equation $\frac{81^{x(x+2)}}{3^{5 x-7}}=9^{x-8}$ . She is able to simplify the equation to the form $4 x^{2}+b x+c=0$ . the value of $c$ is $-15$ 1 23 9

1. We are given an equation involving exponents and need to simplify it to a quadratic form.
2. The bases of the exponents can be rewritten as powers of 3.
3. The simplified quadratic equation is in the form $4x^2 + bx + c = 0$ .

1. Rewrite all terms with a common base.
2. Simplify the equation using properties of exponents.
3. Rearrange the equation into a quadratic form.
4. Identify the value of $c$ .

Apply the properties of exponents to simplify:
$3^{4x(x+2) - (5x-7)} = 3^{2(x-8)}$
This simplifies to:
$4x(x+2) - 5x + 7 = 2(x-8)$

Identify the value of $c$ in the quadratic equation $4x^2 + x + 23 = 0$ .
The value of $c$ is:
$\boxed{23}$