Question Solve the quadratic equation $x^{2} + 8x = 9$ for the value of $x$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the quadratic equation $x^{2}+8x=9$ .
2. We need to solve for the variable $x$ .

STEP 2

To solve the quadratic equation, we first want to set it to the standard form $ax^{2} + bx + c = 0$ .
$x^{2} + 8x - 9 = 0$

STEP 3

Next, we attempt to factor the quadratic equation if possible. We look for two numbers that multiply to give the product of $a \times c$ (which is $1 \times -9 = -9$ ) and add up to $b$ (which is $8$ ).

STEP 4

The two numbers that satisfy these conditions are $9$ and $-1$ because $9 \times (-1) = -9$ and $9 + (-1) = 8$ .

STEP 5

We rewrite the middle term $8x$ using the numbers $9$ and $-1$ to split it into two terms.
$x^{2} + 9x - x - 9 = 0$

STEP 6

Next, we factor by grouping. We group the terms as follows:
$(x^{2} + 9x) - (x + 9) = 0$

STEP 7

We factor out the common factor from each group.
$x(x + 9) - 1(x + 9) = 0$

STEP 8

Now we can see that $x + 9$ is a common factor, so we factor it out.
$(x - 1)(x + 9) = 0$

STEP 9

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero.
So, we set each factor equal to zero:
$x - 1 = 0 \quad \text{or} \quad x + 9 = 0$

STEP 10

Solve each equation for $x$ .
For $x - 1 = 0$ :
$x = 1$
For $x + 9 = 0$ :
$x = -9$

STEP 11

The solutions to the quadratic equation $x^{2} + 8x = 9$ are $x = 1$ and $x = -9$ .

Was this helpful?

Question Solve the quadratic equation $x^{2} + 8x = 9$ for the value of $x$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the quadratic equation $x^{2}+8x=9$ .
2. We need to solve for the variable $x$ .

STEP 2

To solve the quadratic equation, we first want to set it to the standard form $ax^{2} + bx + c = 0$ .
$x^{2} + 8x - 9 = 0$

STEP 3

Next, we attempt to factor the quadratic equation if possible. We look for two numbers that multiply to give the product of $a \times c$ (which is $1 \times -9 = -9$ ) and add up to $b$ (which is $8$ ).

STEP 4

The two numbers that satisfy these conditions are $9$ and $-1$ because $9 \times (-1) = -9$ and $9 + (-1) = 8$ .

STEP 5

We rewrite the middle term $8x$ using the numbers $9$ and $-1$ to split it into two terms.
$x^{2} + 9x - x - 9 = 0$

STEP 6

Next, we factor by grouping. We group the terms as follows:
$(x^{2} + 9x) - (x + 9) = 0$

STEP 7

We factor out the common factor from each group.
$x(x + 9) - 1(x + 9) = 0$

STEP 8

Now we can see that $x + 9$ is a common factor, so we factor it out.
$(x - 1)(x + 9) = 0$

STEP 9

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero.
So, we set each factor equal to zero:
$x - 1 = 0 \quad \text{or} \quad x + 9 = 0$

STEP 10

Solve each equation for $x$ .
For $x - 1 = 0$ :
$x = 1$
For $x + 9 = 0$ :
$x = -9$

STEP 11

The solutions to the quadratic equation $x^{2} + 8x = 9$ are $x = 1$ and $x = -9$ .

Get Studdy

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.

Question Solve the quadratic equation x2+8x=9x^{2} + 8x = 9x2+8x=9 for the value of xxx.

Studdy Solution

STEP 1

Assumptions1. We are given the quadratic equation x2+8x=9x^{2}+8x=9x2+8x=9.2. We need to solve for the variable xxx.

STEP 2

To solve the quadratic equation, we first want to set it to the standard form ax2+bx+c=0ax^{2} + bx + c = 0ax2+bx+c=0.x2+8x−9=0x^{2} + 8x - 9 = 0x2+8x−9=0

STEP 3

Next, we attempt to factor the quadratic equation if possible. We look for two numbers that multiply to give the product of a×ca \times ca×c (which is 1×−9=−91 \times -9 = -91×−9=−9) and add up to bbb (which is 888).

STEP 4

The two numbers that satisfy these conditions are 999 and −1-1−1 because 9×(−1)=−99 \times (-1) = -99×(−1)=−9 and 9+(−1)=89 + (-1) = 89+(−1)=8.

STEP 5

We rewrite the middle term 8x8x8x using the numbers 999 and −1-1−1 to split it into two terms.x2+9x−x−9=0x^{2} + 9x - x - 9 = 0x2+9x−x−9=0

STEP 6

Next, we factor by grouping. We group the terms as follows:(x2+9x)−(x+9)=0(x^{2} + 9x) - (x + 9) = 0(x2+9x)−(x+9)=0

STEP 7

We factor out the common factor from each group.x(x+9)−1(x+9)=0x(x + 9) - 1(x + 9) = 0x(x+9)−1(x+9)=0

STEP 8

Now we can see that x+9x + 9x+9 is a common factor, so we factor it out.(x−1)(x+9)=0(x - 1)(x + 9) = 0(x−1)(x+9)=0

STEP 9

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero.So, we set each factor equal to zero:x−1=0orx+9=0x - 1 = 0 \quad \text{or} \quad x + 9 = 0x−1=0orx+9=0

STEP 10

Solve each equation for xxx.For x−1=0x - 1 = 0x−1=0:x=1x = 1x=1For x+9=0x + 9 = 0x+9=0:x=−9x = -9x=−9

STEP 11

The solutions to the quadratic equation x2+8x=9x^{2} + 8x = 9x2+8x=9 are x=1x = 1x=1 and x=−9x = -9x=−9.

Get Studdy

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.

Question Solve the quadratic equation $x^{2} + 8x = 9$ for the value of $x$ .

Assumptions
1. We are given the quadratic equation $x^{2}+8x=9$ .
2. We need to solve for the variable $x$ .

To solve the quadratic equation, we first want to set it to the standard form $ax^{2} + bx + c = 0$ .
$x^{2} + 8x - 9 = 0$

Next, we attempt to factor the quadratic equation if possible. We look for two numbers that multiply to give the product of $a \times c$ (which is $1 \times -9 = -9$ ) and add up to $b$ (which is $8$ ).

The two numbers that satisfy these conditions are $9$ and $-1$ because $9 \times (-1) = -9$ and $9 + (-1) = 8$ .

We rewrite the middle term $8x$ using the numbers $9$ and $-1$ to split it into two terms.
$x^{2} + 9x - x - 9 = 0$

Next, we factor by grouping. We group the terms as follows:
$(x^{2} + 9x) - (x + 9) = 0$

We factor out the common factor from each group.
$x(x + 9) - 1(x + 9) = 0$

Now we can see that $x + 9$ is a common factor, so we factor it out.
$(x - 1)(x + 9) = 0$

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero.
So, we set each factor equal to zero:
$x - 1 = 0 \quad \text{or} \quad x + 9 = 0$

Solve each equation for $x$ .
For $x - 1 = 0$ :
$x = 1$
For $x + 9 = 0$ :
$x = -9$

The solutions to the quadratic equation $x^{2} + 8x = 9$ are $x = 1$ and $x = -9$ .