Solved on Feb 22, 2024

Solve the quadratic equation $x^{2} + x - 61 = -5$ for the real-valued solutions.

STEP 1

Assumptions
1. We are given a quadratic equation $x^{2} + x - 61 = -5$ .
2. We need to solve for $x$ .

STEP 2

First, we need to simplify the equation by moving all terms to one side to set the equation to zero.
$x^{2} + x - 61 + 5 = 0$

STEP 3

Combine like terms to simplify the equation further.
$x^{2} + x - 56 = 0$

STEP 4

Now we have a quadratic equation in standard form. We will attempt to factor the quadratic, looking for two numbers that multiply to $-56$ and add to $1$ .

STEP 5

Find two numbers that meet the criteria from STEP_4. These numbers are $8$ and $-7$ because $8 \cdot (-7) = -56$ and $8 + (-7) = 1$ .

STEP 6

Rewrite the quadratic equation using the two numbers found in STEP_5 to split the middle term.
$x^{2} + 8x - 7x - 56 = 0$

STEP 7

Factor by grouping. Group the first two terms together and the last two terms together.
$(x^{2} + 8x) - (7x + 56) = 0$

STEP 8

Factor out the greatest common factor from each group.
$x(x + 8) - 7(x + 8) = 0$

STEP 9

Now that we have a common factor of $(x + 8)$ , we can factor it out.
$(x - 7)(x + 8) = 0$

STEP 10

Apply the zero-product property, which states that if a product of two factors is zero, then at least one of the factors must be zero.
$x - 7 = 0 \quad \text{or} \quad x + 8 = 0$

STEP 11

Solve each equation for $x$ .
For the first equation:
$x - 7 = 0$
$x = 7$
For the second equation:
$x + 8 = 0$
$x = -8$

STEP 12

We have found the two solutions to the quadratic equation.
The solutions are $x = 7$ and $x = -8$ .

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Solve the quadratic equation $x^{2} + x - 61 = -5$ for the real-valued solutions.

STEP 1

Assumptions
1. We are given a quadratic equation $x^{2} + x - 61 = -5$ .
2. We need to solve for $x$ .

STEP 2

First, we need to simplify the equation by moving all terms to one side to set the equation to zero.
$x^{2} + x - 61 + 5 = 0$

STEP 3

Combine like terms to simplify the equation further.
$x^{2} + x - 56 = 0$

STEP 4

Now we have a quadratic equation in standard form. We will attempt to factor the quadratic, looking for two numbers that multiply to $-56$ and add to $1$ .

STEP 5

Find two numbers that meet the criteria from STEP_4. These numbers are $8$ and $-7$ because $8 \cdot (-7) = -56$ and $8 + (-7) = 1$ .

STEP 6

Rewrite the quadratic equation using the two numbers found in STEP_5 to split the middle term.
$x^{2} + 8x - 7x - 56 = 0$

STEP 7

Factor by grouping. Group the first two terms together and the last two terms together.
$(x^{2} + 8x) - (7x + 56) = 0$

STEP 8

Factor out the greatest common factor from each group.
$x(x + 8) - 7(x + 8) = 0$

STEP 9

Now that we have a common factor of $(x + 8)$ , we can factor it out.
$(x - 7)(x + 8) = 0$

STEP 10

Apply the zero-product property, which states that if a product of two factors is zero, then at least one of the factors must be zero.
$x - 7 = 0 \quad \text{or} \quad x + 8 = 0$

STEP 11

Solve each equation for $x$ .
For the first equation:
$x - 7 = 0$
$x = 7$
For the second equation:
$x + 8 = 0$
$x = -8$

STEP 12

We have found the two solutions to the quadratic equation.
The solutions are $x = 7$ and $x = -8$ .

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Solve the quadratic equation x2+x−61=−5x^{2} + x - 61 = -5x2+x−61=−5 for the real-valued solutions.

STEP 1

Assumptions1. We are given a quadratic equation x2+x−61=−5x^{2} + x - 61 = -5x2+x−61=−5.2. We need to solve for xxx.

STEP 2

First, we need to simplify the equation by moving all terms to one side to set the equation to zero.x2+x−61+5=0x^{2} + x - 61 + 5 = 0x2+x−61+5=0

STEP 3

Combine like terms to simplify the equation further.x2+x−56=0x^{2} + x - 56 = 0x2+x−56=0

STEP 4

Now we have a quadratic equation in standard form. We will attempt to factor the quadratic, looking for two numbers that multiply to −56-56−56 and add to 111.

STEP 5

Find two numbers that meet the criteria from STEP_4. These numbers are 888 and −7-7−7 because 8⋅(−7)=−568 \cdot (-7) = -568⋅(−7)=−56 and 8+(−7)=18 + (-7) = 18+(−7)=1.

STEP 6

Rewrite the quadratic equation using the two numbers found in STEP_5 to split the middle term.x2+8x−7x−56=0x^{2} + 8x - 7x - 56 = 0x2+8x−7x−56=0

STEP 7

Factor by grouping. Group the first two terms together and the last two terms together.(x2+8x)−(7x+56)=0(x^{2} + 8x) - (7x + 56) = 0(x2+8x)−(7x+56)=0

STEP 8

Factor out the greatest common factor from each group.x(x+8)−7(x+8)=0x(x + 8) - 7(x + 8) = 0x(x+8)−7(x+8)=0

STEP 9

Now that we have a common factor of (x+8)(x + 8)(x+8), we can factor it out.(x−7)(x+8)=0(x - 7)(x + 8) = 0(x−7)(x+8)=0

STEP 10

Apply the zero-product property, which states that if a product of two factors is zero, then at least one of the factors must be zero.x−7=0orx+8=0x - 7 = 0 \quad \text{or} \quad x + 8 = 0x−7=0orx+8=0

STEP 11

Solve each equation for xxx.For the first equation:x−7=0x - 7 = 0x−7=0x=7x = 7x=7For the second equation:x+8=0x + 8 = 0x+8=0x=−8x = -8x=−8

STEP 12

We have found the two solutions to the quadratic equation.The solutions are x=7x = 7x=7 and x=−8x = -8x=−8.

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

Solve the quadratic equation $x^{2} + x - 61 = -5$ for the real-valued solutions.

Assumptions
1. We are given a quadratic equation $x^{2} + x - 61 = -5$ .
2. We need to solve for $x$ .

First, we need to simplify the equation by moving all terms to one side to set the equation to zero.
$x^{2} + x - 61 + 5 = 0$

Combine like terms to simplify the equation further.
$x^{2} + x - 56 = 0$

Now we have a quadratic equation in standard form. We will attempt to factor the quadratic, looking for two numbers that multiply to $-56$ and add to $1$ .

Find two numbers that meet the criteria from STEP_4. These numbers are $8$ and $-7$ because $8 \cdot (-7) = -56$ and $8 + (-7) = 1$ .

Rewrite the quadratic equation using the two numbers found in STEP_5 to split the middle term.
$x^{2} + 8x - 7x - 56 = 0$

Factor by grouping. Group the first two terms together and the last two terms together.
$(x^{2} + 8x) - (7x + 56) = 0$

Factor out the greatest common factor from each group.
$x(x + 8) - 7(x + 8) = 0$

Now that we have a common factor of $(x + 8)$ , we can factor it out.
$(x - 7)(x + 8) = 0$

Apply the zero-product property, which states that if a product of two factors is zero, then at least one of the factors must be zero.
$x - 7 = 0 \quad \text{or} \quad x + 8 = 0$

Solve each equation for $x$ .
For the first equation:
$x - 7 = 0$
$x = 7$
For the second equation:
$x + 8 = 0$
$x = -8$

We have found the two solutions to the quadratic equation.
The solutions are $x = 7$ and $x = -8$ .