Solved on Sep 27, 2023

Rewrite $7 x^{4} = 7000 x$ in factored form. Find the solution set. Choose A) The solution set is $x = 0, x = 10, x = -10 + 10i, x = -10 - 10i$ or B) No solution.

STEP 1

Assumptions1. The equation is $7x^{4} =7000x$ . We are asked to rewrite the equation in factored form and find the solution set

STEP 2

First, we need to rewrite the equation in factored form. The given factored form is $7x(x-10)(x^{2}+10x+100)=0$ .

STEP 3

Now, we need to find the roots of the equation. The roots of the equation are the values of $x$ that make the equation equal to zero.

STEP 4

Set each factor equal to zero and solve for $x$ .
$7x =0$ $x -10 =0$ $x^{2} +10x +100 =0$

STEP 5

olve the first equation for $x$ .
$7x =0$ $x =0$

STEP 6

olve the second equation for $x$ .
$x -10 =0$ $x =10$

STEP 7

The third equation is a quadratic equation. We can solve it using the quadratic formula $x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$ .

STEP 8

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation $x^{2} +10x +100 =0$ .
$a =1$ $b =10$ $c =100$

STEP 9

Substitute the coefficients into the quadratic formula.
$x = \frac{- \pm \sqrt{^{2}-4()(100)}}{2()}$

STEP 10

implify the expression under the square root.
$x = \frac{-10 \pm \sqrt{100-400}}{2}$

STEP 11

implify further.
$x = \frac{-10 \pm \sqrt{-300}}{}$

STEP 12

Since the value under the square root is negative, we have complex roots. We can write the square root of -300 as $i\sqrt{300}$ .
$x = \frac{-10 \pm i\sqrt{300}}{2}$

STEP 13

implify the expression.
$x = -5 \pm5i\sqrt{3}$ So, the solutions to the equation are $x =0$ , $x =10$ , and $x = -5 \pm5i\sqrt{3}$ . Therefore, the solution set is $\{0,10, -5 +5i\sqrt{3}, -5 -5i\sqrt{3}\}$ .

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Rewrite $7 x^{4} = 7000 x$ in factored form. Find the solution set. Choose A) The solution set is $x = 0, x = 10, x = -10 + 10i, x = -10 - 10i$ or B) No solution.

STEP 1

Assumptions1. The equation is $7x^{4} =7000x$ . We are asked to rewrite the equation in factored form and find the solution set

STEP 2

First, we need to rewrite the equation in factored form. The given factored form is $7x(x-10)(x^{2}+10x+100)=0$ .

STEP 3

Now, we need to find the roots of the equation. The roots of the equation are the values of $x$ that make the equation equal to zero.

STEP 4

Set each factor equal to zero and solve for $x$ .
$7x =0$ $x -10 =0$ $x^{2} +10x +100 =0$

STEP 5

olve the first equation for $x$ .
$7x =0$ $x =0$

STEP 6

olve the second equation for $x$ .
$x -10 =0$ $x =10$

STEP 7

The third equation is a quadratic equation. We can solve it using the quadratic formula $x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$ .

STEP 8

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation $x^{2} +10x +100 =0$ .
$a =1$ $b =10$ $c =100$

STEP 9

Substitute the coefficients into the quadratic formula.
$x = \frac{- \pm \sqrt{^{2}-4()(100)}}{2()}$

STEP 10

implify the expression under the square root.
$x = \frac{-10 \pm \sqrt{100-400}}{2}$

STEP 11

implify further.
$x = \frac{-10 \pm \sqrt{-300}}{}$

STEP 12

Since the value under the square root is negative, we have complex roots. We can write the square root of -300 as $i\sqrt{300}$ .
$x = \frac{-10 \pm i\sqrt{300}}{2}$

STEP 13

implify the expression.
$x = -5 \pm5i\sqrt{3}$ So, the solutions to the equation are $x =0$ , $x =10$ , and $x = -5 \pm5i\sqrt{3}$ . Therefore, the solution set is $\{0,10, -5 +5i\sqrt{3}, -5 -5i\sqrt{3}\}$ .

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Rewrite 7x4=7000x7 x^{4} = 7000 x7x4=7000x in factored form. Find the solution set. Choose A) The solution set is x=0,x=10,x=−10+10i,x=−10−10ix = 0, x = 10, x = -10 + 10i, x = -10 - 10ix=0,x=10,x=−10+10i,x=−10−10i or B) No solution.

STEP 1

Assumptions1. The equation is 7x4=7000x7x^{4} =7000x7x4=7000x . We are asked to rewrite the equation in factored form and find the solution set

STEP 2

First, we need to rewrite the equation in factored form. The given factored form is 7x(x−10)(x2+10x+100)=07x(x-10)(x^{2}+10x+100)=07x(x−10)(x2+10x+100)=0.

STEP 3

Now, we need to find the roots of the equation. The roots of the equation are the values of xxx that make the equation equal to zero.

STEP 4

Set each factor equal to zero and solve for xxx.7x=07x =07x=0x−10=0x -10 =0x−10=0x2+10x+100=0x^{2} +10x +100 =0x2+10x+100=0

STEP 5

olve the first equation for xxx.7x=07x =07x=0x=0x =0x=0

STEP 6

olve the second equation for xxx.x−10=0x -10 =0x−10=0x=10x =10x=10

STEP 7

The third equation is a quadratic equation. We can solve it using the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}x=2a−b±b2−4ac​​.

STEP 8

Identify the coefficients aaa, bbb, and ccc from the quadratic equation x2+10x+100=0x^{2} +10x +100 =0x2+10x+100=0.a=1a =1a=1b=10b =10b=10c=100c =100c=100

STEP 9

Substitute the coefficients into the quadratic formula.x=−±2−4()(100)2()x = \frac{- \pm \sqrt{^{2}-4()(100)}}{2()}x=2()−±2−4()(100)​​

STEP 10

implify the expression under the square root.x=−10±100−4002x = \frac{-10 \pm \sqrt{100-400}}{2}x=2−10±100−400​​

STEP 11

implify further.x=−10±−300x = \frac{-10 \pm \sqrt{-300}}{}x=−10±−300​​

STEP 12

Since the value under the square root is negative, we have complex roots. We can write the square root of -300 as i300i\sqrt{300}i300​.x=−10±i3002x = \frac{-10 \pm i\sqrt{300}}{2}x=2−10±i300​​

STEP 13

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Rewrite $7 x^{4} = 7000 x$ in factored form. Find the solution set. Choose A) The solution set is $x = 0, x = 10, x = -10 + 10i, x = -10 - 10i$ or B) No solution.

Assumptions1. The equation is $7x^{4} =7000x$ . We are asked to rewrite the equation in factored form and find the solution set

First, we need to rewrite the equation in factored form. The given factored form is $7x(x-10)(x^{2}+10x+100)=0$ .

Now, we need to find the roots of the equation. The roots of the equation are the values of $x$ that make the equation equal to zero.

Set each factor equal to zero and solve for $x$ .
$7x =0$ $x -10 =0$ $x^{2} +10x +100 =0$

olve the first equation for $x$ .
$7x =0$ $x =0$

olve the second equation for $x$ .
$x -10 =0$ $x =10$

The third equation is a quadratic equation. We can solve it using the quadratic formula $x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$ .

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation $x^{2} +10x +100 =0$ .
$a =1$ $b =10$ $c =100$

Substitute the coefficients into the quadratic formula.
$x = \frac{- \pm \sqrt{^{2}-4()(100)}}{2()}$

implify the expression under the square root.
$x = \frac{-10 \pm \sqrt{100-400}}{2}$

implify further.
$x = \frac{-10 \pm \sqrt{-300}}{}$

Since the value under the square root is negative, we have complex roots. We can write the square root of -300 as $i\sqrt{300}$ .
$x = \frac{-10 \pm i\sqrt{300}}{2}$