Solve for x 0.8(x^2+2x+110)^(2/3)=20

$0.8 {(x^{2} + 2 x + 110)}^{\frac{2}{3}} = 20$

Step 1

Divide each term in $0.8 {(x^{2} + 2 x + 110)}^{\frac{2}{3}} = 20$ by $0.8$ and simplify.

Divide each term in $0.8 {(x^{2} + 2 x + 110)}^{\frac{2}{3}} = 20$ by $0.8$ .

$\frac{0.8 {(x^{2} + 2 x + 110)}^{\frac{2}{3}}}{0.8} = \frac{20}{0.8}$

Simplify the left side.

Cancel the common factor.

$\frac{0.8 {(x^{2} + 2 x + 110)}^{\frac{2}{3}}}{0.8} = \frac{20}{0.8}$

Divide ${(x^{2} + 2 x + 110)}^{\frac{2}{3}}$ by $1$ .

${(x^{2} + 2 x + 110)}^{\frac{2}{3}} = \frac{20}{0.8}$

Simplify the right side.

Divide $20$ by $0.8$ .

${(x^{2} + 2 x + 110)}^{\frac{2}{3}} = 25$

Step 2

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${({(x^{2} + 2 x + 110)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 25^{\frac{3}{2}}$

Step 3

Simplify the exponent.

Simplify the left side.

Simplify ${({(x^{2} + 2 x + 110)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Multiply the exponents in ${({(x^{2} + 2 x + 110)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} + 2 x + 110)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 25^{\frac{3}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} + 2 x + 110)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 25^{\frac{3}{2}}$

Rewrite the expression.

${(x^{2} + 2 x + 110)}^{\frac{1}{3} \cdot 3} = \pm 25^{\frac{3}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x^{2} + 2 x + 110)}^{\frac{1}{3} \cdot 3} = \pm 25^{\frac{3}{2}}$

Rewrite the expression.

${(x^{2} + 2 x + 110)}^{1} = \pm 25^{\frac{3}{2}}$

Simplify.

$x^{2} + 2 x + 110 = \pm 25^{\frac{3}{2}}$

Simplify the right side.

Simplify $\pm 25^{\frac{3}{2}}$ .

Simplify the expression.

Rewrite $25$ as $5^{2}$ .

$x^{2} + 2 x + 110 = \pm {(5^{2})}^{\frac{3}{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{2} + 2 x + 110 = \pm 5^{2 (\frac{3}{2})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{2} + 2 x + 110 = \pm 5^{2 (\frac{3}{2})}$

Rewrite the expression.

$x^{2} + 2 x + 110 = \pm 5^{3}$

Raise $5$ to the power of $3$ .

$x^{2} + 2 x + 110 = \pm 125$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x^{2} + 2 x + 110 = 125$

Subtract $125$ from both sides of the equation.

$x^{2} + 2 x + 110 - 125 = 0$

Subtract $125$ from $110$ .

$x^{2} + 2 x - 15 = 0$

Factor $x^{2} + 2 x - 15$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 15$ and whose sum is $2$ .

$- 3, 5$

Write the factored form using these integers.

$(x - 3) (x + 5) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 3 = 0$

$x + 5 = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

The final solution is all the values that make $(x - 3) (x + 5) = 0$ true.

$x = 3, - 5$

Next, use the negative value of the $\pm$ to find the second solution.

$x^{2} + 2 x + 110 = - 125$

Move all terms to the left side of the equation and simplify.

Add $125$ to both sides of the equation.

$x^{2} + 2 x + 110 + 125 = 0$

Add $110$ and $125$ .

$x^{2} + 2 x + 235 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 2$ , and $c = 235$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 235)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $2$ to the power of $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 235}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 235$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 235}}{2 \cdot 1}$

Multiply $- 4$ by $235$ .

$x = \frac{- 2 \pm \sqrt{4 - 940}}{2 \cdot 1}$

Subtract $940$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 936}}{2 \cdot 1}$

Rewrite $- 936$ as $- 1 (936)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 936}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (936)}$ as $\sqrt{- 1} \cdot \sqrt{936}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{936}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{936}}{2 \cdot 1}$

Rewrite $936$ as $6^{2} \cdot 26$ .

Factor $36$ out of $936$ .

$x = \frac{- 2 \pm i \cdot \sqrt{36 (26)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{6^{2} \cdot 26}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (6 \sqrt{26})}{2 \cdot 1}$

Move $6$ to the left of $i$ .

$x = \frac{- 2 \pm 6 i \sqrt{26}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 6 i \sqrt{26}}{2}$

Simplify $\frac{- 2 \pm 6 i \sqrt{26}}{2}$ .

$x = - 1 \pm 3 i \sqrt{26}$

The final answer is the combination of both solutions.

$x = - 1 + 3 i \sqrt{26}, - 1 - 3 i \sqrt{26}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 5, - 1 + 3 i \sqrt{26}, - 1 - 3 i \sqrt{26}$

Solve for x 0.8(x^2+2x+110)^(2/3)=20

Related Questions

Support LCMGCF.com by disabling your adblocker.