Solved on Dec 14, 2023

Find the equation equivalent to $0=-3 x^{2}+27 x+30$ .

STEP 1

Assumptions
1. We are given the quadratic equation $0 = -3x^2 + 27x + 30$ .
2. We need to find an equivalent equation, which means it has the same solutions as the given equation.

STEP 2

To find an equivalent equation, we can factor out the greatest common factor (GCF) from all terms of the given equation.

STEP 3

Identify the GCF of the coefficients of the terms in the equation.
The GCF of $-3, 27,$ and $30$ is $3$ .

STEP 4

Factor out the GCF from each term of the equation.
$0 = -3(x^2 - 9x - 10)$

STEP 5

Now, we need to factor the quadratic expression inside the parentheses.

STEP 6

To factor the quadratic expression $x^2 - 9x - 10$ , we look for two numbers that multiply to $-10$ (the constant term) and add up to $-9$ (the coefficient of the linear term).

STEP 7

The two numbers that satisfy these conditions are $-10$ and $1$ , because $-10 \times 1 = -10$ and $-10 + 1 = -9$ .

STEP 8

Use these two numbers to express the quadratic expression as a product of two binomials.
$x^2 - 9x - 10 = (x - 10)(x + 1)$

STEP 9

Substitute the factored form of the quadratic expression back into the equation.
$0 = -3(x - 10)(x + 1)$

STEP 10

The equation $0 = -3(x - 10)(x + 1)$ is equivalent to the given equation $0 = -3x^2 + 27x + 30$ .
The equivalent equation is $0 = -3(x - 10)(x + 1)$ .

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Find the equation equivalent to $0=-3 x^{2}+27 x+30$ .

STEP 1

Assumptions
1. We are given the quadratic equation $0 = -3x^2 + 27x + 30$ .
2. We need to find an equivalent equation, which means it has the same solutions as the given equation.

STEP 2

To find an equivalent equation, we can factor out the greatest common factor (GCF) from all terms of the given equation.

STEP 3

Identify the GCF of the coefficients of the terms in the equation.
The GCF of $-3, 27,$ and $30$ is $3$ .

STEP 4

Factor out the GCF from each term of the equation.
$0 = -3(x^2 - 9x - 10)$

STEP 5

Now, we need to factor the quadratic expression inside the parentheses.

STEP 6

To factor the quadratic expression $x^2 - 9x - 10$ , we look for two numbers that multiply to $-10$ (the constant term) and add up to $-9$ (the coefficient of the linear term).

STEP 7

The two numbers that satisfy these conditions are $-10$ and $1$ , because $-10 \times 1 = -10$ and $-10 + 1 = -9$ .

STEP 8

Use these two numbers to express the quadratic expression as a product of two binomials.
$x^2 - 9x - 10 = (x - 10)(x + 1)$

STEP 9

Substitute the factored form of the quadratic expression back into the equation.
$0 = -3(x - 10)(x + 1)$

STEP 10

The equation $0 = -3(x - 10)(x + 1)$ is equivalent to the given equation $0 = -3x^2 + 27x + 30$ .
The equivalent equation is $0 = -3(x - 10)(x + 1)$ .

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Find the equation equivalent to 0=−3x2+27x+300=-3 x^{2}+27 x+300=−3x2+27x+30.

STEP 1

Assumptions1. We are given the quadratic equation 0=−3x2+27x+300 = -3x^2 + 27x + 300=−3x2+27x+30.2. We need to find an equivalent equation, which means it has the same solutions as the given equation.

STEP 2

To find an equivalent equation, we can factor out the greatest common factor (GCF) from all terms of the given equation.

STEP 3

Identify the GCF of the coefficients of the terms in the equation.The GCF of −3,27, -3, 27, −3,27, and 30 30 30 is 3 3 3.

STEP 4

Factor out the GCF from each term of the equation.0=−3(x2−9x−10)0 = -3(x^2 - 9x - 10)0=−3(x2−9x−10)

STEP 5

Now, we need to factor the quadratic expression inside the parentheses.

STEP 6

To factor the quadratic expression x2−9x−10x^2 - 9x - 10x2−9x−10, we look for two numbers that multiply to −10 -10 −10 (the constant term) and add up to −9 -9 −9 (the coefficient of the linear term).

STEP 7

The two numbers that satisfy these conditions are −10 -10 −10 and 1 1 1, because −10×1=−10 -10 \times 1 = -10 −10×1=−10 and −10+1=−9 -10 + 1 = -9 −10+1=−9.

STEP 8

Use these two numbers to express the quadratic expression as a product of two binomials.x2−9x−10=(x−10)(x+1)x^2 - 9x - 10 = (x - 10)(x + 1)x2−9x−10=(x−10)(x+1)

STEP 9

Substitute the factored form of the quadratic expression back into the equation.0=−3(x−10)(x+1)0 = -3(x - 10)(x + 1)0=−3(x−10)(x+1)

STEP 10

The equation 0=−3(x−10)(x+1)0 = -3(x - 10)(x + 1)0=−3(x−10)(x+1) is equivalent to the given equation 0=−3x2+27x+300 = -3x^2 + 27x + 300=−3x2+27x+30.The equivalent equation is 0=−3(x−10)(x+1)0 = -3(x - 10)(x + 1)0=−3(x−10)(x+1).

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

Find the equation equivalent to $0=-3 x^{2}+27 x+30$ .

Assumptions
1. We are given the quadratic equation $0 = -3x^2 + 27x + 30$ .
2. We need to find an equivalent equation, which means it has the same solutions as the given equation.

Identify the GCF of the coefficients of the terms in the equation.
The GCF of $-3, 27,$ and $30$ is $3$ .

Factor out the GCF from each term of the equation.
$0 = -3(x^2 - 9x - 10)$

To factor the quadratic expression $x^2 - 9x - 10$ , we look for two numbers that multiply to $-10$ (the constant term) and add up to $-9$ (the coefficient of the linear term).

The two numbers that satisfy these conditions are $-10$ and $1$ , because $-10 \times 1 = -10$ and $-10 + 1 = -9$ .

Use these two numbers to express the quadratic expression as a product of two binomials.
$x^2 - 9x - 10 = (x - 10)(x + 1)$

Substitute the factored form of the quadratic expression back into the equation.
$0 = -3(x - 10)(x + 1)$

The equation $0 = -3(x - 10)(x + 1)$ is equivalent to the given equation $0 = -3x^2 + 27x + 30$ .
The equivalent equation is $0 = -3(x - 10)(x + 1)$ .