Solve for w 4w^4+40w^2-44=0

$4 w^{4} + 40 w^{2} - 44 = 0$

Substitute $u = w^{2}$ into the equation. This will make the quadratic formula easy to use.

$4 u^{2} + 40 u - 44 = 0$

$u = w^{2}$

Factor the left side of the equation.

Tap for more steps…

Factor $4$ out of $4 u^{2} + 40 u - 44$ .

Tap for more steps…

Factor $4$ out of $4 u^{2}$ .

$4 (u^{2}) + 40 u - 44 = 0$

Factor $4$ out of $40 u$ .

$4 (u^{2}) + 4 (10 u) - 44 = 0$

Factor $4$ out of $- 44$ .

$4 u^{2} + 4 (10 u) + 4 \cdot - 11 = 0$

Factor $4$ out of $4 u^{2} + 4 (10 u)$ .

$4 (u^{2} + 10 u) + 4 \cdot - 11 = 0$

Factor $4$ out of $4 (u^{2} + 10 u) + 4 \cdot - 11$ .

$4 (u^{2} + 10 u - 11) = 0$

Factor.

Tap for more steps…

Factor $u^{2} + 10 u - 11$ using the AC method.

Tap for more steps…

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 $- 11$ and whose sum is $10$ .

$- 1, 11$

Write the factored form using these integers.

$4 ((u - 1) (u + 11)) = 0$

Remove unnecessary parentheses.

$4 (u - 1) (u + 11) = 0$

Divide each term by $4$ and simplify.

Tap for more steps…

Divide each term in $4 (u - 1) (u + 11) = 0$ by $4$ .

$\frac{4 (u - 1) (u + 11)}{4} = \frac{0}{4}$

Simplify $\frac{4 (u - 1) (u + 11)}{4}$ .

Tap for more steps…

Cancel the common factor of $4$ .

Tap for more steps…

Cancel the common factor.

$\frac{4 (u - 1) (u + 11)}{4} = \frac{0}{4}$

Divide $(u - 1) (u + 11)$ by $1$ .

$(u - 1) (u + 11) = \frac{0}{4}$

Expand $(u - 1) (u + 11)$ using the FOIL Method.

Tap for more steps…

Apply the distributive property.

$u (u + 11) - 1 (u + 11) = \frac{0}{4}$

Apply the distributive property.

$u \cdot u + u \cdot 11 - 1 (u + 11) = \frac{0}{4}$

Apply the distributive property.

$u \cdot u + u \cdot 11 - 1 u - 1 \cdot 11 = \frac{0}{4}$

Simplify and combine like terms.

Tap for more steps…

Simplify each term.

Tap for more steps…

Multiply $u$ by $u$ .

$u^{2} + u \cdot 11 - 1 u - 1 \cdot 11 = \frac{0}{4}$

Move $11$ to the left of $u$ .

$u^{2} + 11 \cdot u - 1 u - 1 \cdot 11 = \frac{0}{4}$

Rewrite $- 1 u$ as $- u$ .

$u^{2} + 11 u - u - 1 \cdot 11 = \frac{0}{4}$

Multiply $- 1$ by $11$ .

$u^{2} + 11 u - u - 11 = \frac{0}{4}$

Subtract $u$ from $11 u$ .

$u^{2} + 10 u - 11 = \frac{0}{4}$

Divide $0$ by $4$ .

$u^{2} + 10 u - 11 = 0$

Factor $u^{2} + 10 u - 11$ using the AC method.

Tap for more steps…

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 $- 11$ and whose sum is $10$ .

$- 1, 11$

Write the factored form using these integers.

$(u - 1) (u + 11) = 0$

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

$u - 1 = 0$

$u + 11 = 0$

Set the first factor equal to $0$ and solve.

Tap for more steps…

Set the first factor equal to $0$ .

$u - 1 = 0$

Add $1$ to both sides of the equation.

$u = 1$

Set the next factor equal to $0$ and solve.

Tap for more steps…

Set the next factor equal to $0$ .

$u + 11 = 0$

Subtract $11$ from both sides of the equation.

$u = - 11$

The final solution is all the values that make $(u - 1) (u + 11) = 0$ true.

$u = 1, - 11$

Substitute the real value of $u = w^{2}$ back into the solved equation.

$w^{2} = 1$

${(w^{2})}^{1} = - 11$

Solve the first equation for $w$ .

$w^{2} = 1$

Solve the equation for $w$ .

Tap for more steps…

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$w = \pm \sqrt{1}$

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

Tap for more steps…

Any root of $1$ is $1$ .

$w = \pm 1$

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

Tap for more steps…

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

$w = 1$

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

$w = - 1$

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

$w = 1, - 1$

Solve the second equation for $w$ .

${(w^{2})}^{1} = - 11$

Solve the equation for $w$ .

Tap for more steps…

Take the 1th root of each side of the equation to set up the solution for $w$

${(w^{2})}^{1 \cdot \frac{1}{1}} = \sqrt[1]{- 11}$

Remove the perfect root factor $w^{2}$ under the radical to solve for $w$ .

$w^{2} = \sqrt[1]{- 11}$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$w = \pm \sqrt{\sqrt[1]{- 11}}$

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

Tap for more steps…

Simplify the right side of the equation.

Tap for more steps…

Evaluate $\sqrt[1]{- 11}$ as $- 11$ .

$w = \pm \sqrt{- 11}$

Rewrite $- 11$ as $- 1 (11)$ .

$w = \pm \sqrt{- 1 \cdot 11}$

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$w = \pm \sqrt{- 1} \cdot \sqrt{11}$

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

$w = \pm i \sqrt{11}$

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

Tap for more steps…

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

$w = i \sqrt{11}$

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

$w = - i \sqrt{11}$

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

$w = i \sqrt{11}, - i \sqrt{11}$

The solution to $4 w^{4} + 40 w^{2} - 44 = 0$ is $w = 1, - 1, i \sqrt{11}, - i \sqrt{11}$ .

$w = 1, - 1, i \sqrt{11}, - i \sqrt{11}$

Solve for w 4w^4+40w^2-44=0

Solving MATH problems