Solved on Dec 06, 2023

Solve the quadratic equation $x^{2} + 8x = 9$ . Write the expanded and factored forms, a solution, and check your work.

STEP 1

Assumptions
1. We are given the equation $x^{2}+8x=9$ .
2. We need to write the expanded form of the square equation.
3. We need to write the factored form of the square equation.
4. We need to find a solution to the equation.
5. The equation is not a perfect square, which implies we may need to complete the square to transform it into a square equation.

STEP 2

To write the expanded form of the square equation, we need to complete the square. This involves adding and subtracting a certain number to make the left side of the equation a perfect square trinomial.

STEP 3

Identify the coefficient of $x$ which is $8$ and take half of it, then square the result to find the number to add and subtract.
$\left(\frac{8}{2}\right)^2 = 4^2 = 16$

STEP 4

Add and subtract $16$ to the left side of the equation to complete the square.
$x^2 + 8x + 16 - 16 = 9$

STEP 5

Rewrite the equation grouping the perfect square trinomial and moving the constant to the other side of the equation.
$x^2 + 8x + 16 = 9 + 16$

STEP 6

Simplify the right side of the equation.
$x^2 + 8x + 16 = 25$

STEP 7

Now we have the expanded form of the square equation:
$x^2 + 8x + 16 = 25$

STEP 8

The factored form of the square equation is obtained by expressing the perfect square trinomial as the square of a binomial.
$(x + 4)^2 = 25$

STEP 9

To find the solution to the equation, we take the square root of both sides.
$\sqrt{(x + 4)^2} = \pm\sqrt{25}$

STEP 10

Solve for $x$ .
$x + 4 = \pm5$

STEP 11

Find the two solutions by separating the positive and negative cases.
For the positive case:
$x + 4 = 5$
$x = 5 - 4$
$x = 1$
For the negative case:
$x + 4 = -5$
$x = -5 - 4$
$x = -9$

STEP 12

We have found the solutions to the equation, which are $x = 1$ and $x = -9$ .

STEP 13

Fill in the table with the expanded form, the factored form, and the solutions to the equation.
\begin{tabular}{|c|c|c|c|} \hline \begin{tabular}{c} Expanded \\ form of the \\ square \\ equation \end{tabular} & \begin{tabular}{c} factored form \\ of the \\ equation \end{tabular} & \begin{tabular}{c} What is a \\ solution to \\ this equation? \end{tabular} & \begin{tabular}{c} Did you get it \\ right? \end{tabular} \\ \hline $x^2 + 8x + 16 = 25$ & $(x + 4)^2 = 25$ & $x = 1, x = -9$ & Yes \\ \hline \end{tabular}

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Solve the quadratic equation $x^{2} + 8x = 9$ . Write the expanded and factored forms, a solution, and check your work.

STEP 1

STEP 2

To write the expanded form of the square equation, we need to complete the square. This involves adding and subtracting a certain number to make the left side of the equation a perfect square trinomial.

STEP 3

Identify the coefficient of $x$ which is $8$ and take half of it, then square the result to find the number to add and subtract.
$\left(\frac{8}{2}\right)^2 = 4^2 = 16$

STEP 4

Add and subtract $16$ to the left side of the equation to complete the square.
$x^2 + 8x + 16 - 16 = 9$

STEP 5

Rewrite the equation grouping the perfect square trinomial and moving the constant to the other side of the equation.
$x^2 + 8x + 16 = 9 + 16$

STEP 6

Simplify the right side of the equation.
$x^2 + 8x + 16 = 25$

STEP 7

Now we have the expanded form of the square equation:
$x^2 + 8x + 16 = 25$

STEP 8

The factored form of the square equation is obtained by expressing the perfect square trinomial as the square of a binomial.
$(x + 4)^2 = 25$

STEP 9

To find the solution to the equation, we take the square root of both sides.
$\sqrt{(x + 4)^2} = \pm\sqrt{25}$

STEP 10

Solve for $x$ .
$x + 4 = \pm5$

STEP 11

Find the two solutions by separating the positive and negative cases.
For the positive case:
$x + 4 = 5$
$x = 5 - 4$
$x = 1$
For the negative case:
$x + 4 = -5$
$x = -5 - 4$
$x = -9$

STEP 12

We have found the solutions to the equation, which are $x = 1$ and $x = -9$ .

STEP 13

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Solve the quadratic equation x2+8x=9x^{2} + 8x = 9x2+8x=9. Write the expanded and factored forms, a solution, and check your work.

STEP 1

STEP 2

To write the expanded form of the square equation, we need to complete the square. This involves adding and subtracting a certain number to make the left side of the equation a perfect square trinomial.

STEP 3

Identify the coefficient of xxx which is 888 and take half of it, then square the result to find the number to add and subtract.(82)2=42=16\left(\frac{8}{2}\right)^2 = 4^2 = 16(28​)2=42=16

STEP 4

Add and subtract 161616 to the left side of the equation to complete the square.x2+8x+16−16=9x^2 + 8x + 16 - 16 = 9x2+8x+16−16=9

STEP 5

Rewrite the equation grouping the perfect square trinomial and moving the constant to the other side of the equation.x2+8x+16=9+16x^2 + 8x + 16 = 9 + 16x2+8x+16=9+16

STEP 6

Simplify the right side of the equation.x2+8x+16=25x^2 + 8x + 16 = 25x2+8x+16=25

STEP 7

Now we have the expanded form of the square equation:x2+8x+16=25x^2 + 8x + 16 = 25x2+8x+16=25

STEP 8

The factored form of the square equation is obtained by expressing the perfect square trinomial as the square of a binomial.(x+4)2=25(x + 4)^2 = 25(x+4)2=25

STEP 9

To find the solution to the equation, we take the square root of both sides.(x+4)2=±25\sqrt{(x + 4)^2} = \pm\sqrt{25}(x+4)2​=±25​

STEP 10

Solve for xxx.x+4=±5x + 4 = \pm5x+4=±5

STEP 11

Find the two solutions by separating the positive and negative cases.For the positive case:x+4=5x + 4 = 5x+4=5x=5−4x = 5 - 4x=5−4x=1x = 1x=1For the negative case:x+4=−5x + 4 = -5x+4=−5x=−5−4x = -5 - 4x=−5−4x=−9x = -9x=−9

STEP 12

We have found the solutions to the equation, which are x=1x = 1x=1 and x=−9x = -9x=−9.

STEP 13

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Solve the quadratic equation $x^{2} + 8x = 9$ . Write the expanded and factored forms, a solution, and check your work.

Identify the coefficient of $x$ which is $8$ and take half of it, then square the result to find the number to add and subtract.
$\left(\frac{8}{2}\right)^2 = 4^2 = 16$

Add and subtract $16$ to the left side of the equation to complete the square.
$x^2 + 8x + 16 - 16 = 9$

Rewrite the equation grouping the perfect square trinomial and moving the constant to the other side of the equation.
$x^2 + 8x + 16 = 9 + 16$

Simplify the right side of the equation.
$x^2 + 8x + 16 = 25$

Now we have the expanded form of the square equation:
$x^2 + 8x + 16 = 25$

The factored form of the square equation is obtained by expressing the perfect square trinomial as the square of a binomial.
$(x + 4)^2 = 25$

To find the solution to the equation, we take the square root of both sides.
$\sqrt{(x + 4)^2} = \pm\sqrt{25}$

Solve for $x$ .
$x + 4 = \pm5$

Find the two solutions by separating the positive and negative cases.
For the positive case:
$x + 4 = 5$
$x = 5 - 4$
$x = 1$
For the negative case:
$x + 4 = -5$
$x = -5 - 4$
$x = -9$

We have found the solutions to the equation, which are $x = 1$ and $x = -9$ .