Algebra Examples

${(y^{2} - 11)}^{2} - 10 (y^{2} - 11) = - 25$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Apply the distributive property.

${(y^{2} - 11)}^{2} - 10 y^{2} - 10 \cdot - 11 = - 25$

Step 1.2

Multiply $- 10$ by $- 11$ .

${(y^{2} - 11)}^{2} - 10 y^{2} + 110 = - 25$

Step 2

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

$u^{2} - 32 u + 231 = - 25$

$u = y^{2}$

Step 3

Add $25$ to both sides of the equation.

$u^{2} - 32 u + 231 + 25 = 0$

Step 4

Add $231$ and $25$ .

$u^{2} - 32 u + 256 = 0$

Step 5

Factor using the perfect square rule.

Tap for more steps...

Step 5.1

Rewrite $256$ as $16^{2}$ .

$u^{2} - 32 u + 16^{2} = 0$

Step 5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$32 u = 2 \cdot u \cdot 16$

Step 5.3

Rewrite the polynomial.

$u^{2} - 2 \cdot u \cdot 16 + 16^{2} = 0$

Step 5.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 16$ .

${(u - 16)}^{2} = 0$

Step 6

Set the $u - 16$ equal to $0$ .

$u - 16 = 0$

Step 7

Add $16$ to both sides of the equation.

$u = 16$

Step 8

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

$y^{2} = 16$

Step 9

Solve the equation for $y$ .

Tap for more steps...

Step 9.1

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

$y = \pm \sqrt{16}$

Step 9.2

Simplify $\pm \sqrt{16}$ .

Tap for more steps...

Step 9.2.1

Rewrite $16$ as $4^{2}$ .

$y = \pm \sqrt{4^{2}}$

Step 9.2.2

Pull terms out from under the radical, assuming positive real numbers.

$y = \pm 4$

Step 9.3

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

Tap for more steps...

Step 9.3.1

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

$y = 4$

Step 9.3.2

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

$y = - 4$

Step 9.3.3

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

$y = 4, - 4$