Basic Math Examples

$64 y^{2} - y^{4} - 400 = 0$

Step 1

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

$- u^{2} + 64 u - 400 = 0$

$u = y^{2}$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = - 1$ , $b = 64$ , and $c = - 400$ into the quadratic formula and solve for $u$ .

$\frac{- 64 \pm \sqrt{64^{2} - 4 \cdot (- 1 \cdot - 400)}}{2 \cdot - 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $64$ to the power of $2$ .

$u = \frac{- 64 \pm \sqrt{4096 - 4 \cdot - 1 \cdot - 400}}{2 \cdot - 1}$

Step 4.1.2

Multiply $- 4 \cdot - 1 \cdot - 400$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 1$ .

$u = \frac{- 64 \pm \sqrt{4096 + 4 \cdot - 400}}{2 \cdot - 1}$

Step 4.1.2.2

Multiply $4$ by $- 400$ .

$u = \frac{- 64 \pm \sqrt{4096 - 1600}}{2 \cdot - 1}$

Step 4.1.3

Subtract $1600$ from $4096$ .

$u = \frac{- 64 \pm \sqrt{2496}}{2 \cdot - 1}$

Step 4.1.4

Rewrite $2496$ as $8^{2} \cdot 39$ .

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Step 4.1.4.1

Factor $64$ out of $2496$ .

$u = \frac{- 64 \pm \sqrt{64 (39)}}{2 \cdot - 1}$

Step 4.1.4.2

Rewrite $64$ as $8^{2}$ .

$u = \frac{- 64 \pm \sqrt{8^{2} \cdot 39}}{2 \cdot - 1}$

Step 4.1.5

Pull terms out from under the radical.

$u = \frac{- 64 \pm 8 \sqrt{39}}{2 \cdot - 1}$

Step 4.2

Multiply $2$ by $- 1$ .

$u = \frac{- 64 \pm 8 \sqrt{39}}{- 2}$

Step 4.3

Simplify $\frac{- 64 \pm 8 \sqrt{39}}{- 2}$ .

$u = 32 \pm 4 \sqrt{39}$

Step 5

The final answer is the combination of both solutions.

$u = 32 + 4 \sqrt{39}, 32 - 4 \sqrt{39}$

Step 6

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

$y^{2} = 56.97999199$

${(y^{2})}^{1} = 7.020008$

Step 7

Solve the first equation for $y$ .

$y^{2} = 56.97999199$

Step 8

Solve the equation for $y$ .

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Step 8.1

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

$y = \pm \sqrt{56.97999199}$

Step 8.2

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

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Step 8.2.1

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

$y = \sqrt{56.97999199}$

Step 8.2.2

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

$y = - \sqrt{56.97999199}$

Step 8.2.3

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

$y = \sqrt{56.97999199}, - \sqrt{56.97999199}$

Step 9

Solve the second equation for $y$ .

${(y^{2})}^{1} = 7.020008$

Step 10

Solve the equation for $y$ .

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Step 10.1

Remove parentheses.

$y^{2} = 7.020008$

Step 10.2

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

$y = \pm \sqrt{7.020008}$

Step 10.3

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

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Step 10.3.1

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

$y = \sqrt{7.020008}$

Step 10.3.2

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

$y = - \sqrt{7.020008}$

Step 10.3.3

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

$y = \sqrt{7.020008}, - \sqrt{7.020008}$

Step 11

The solution to $- y^{4} + 64 y^{2} - 400 = 0$ is $y = \sqrt{56.97999199}, - \sqrt{56.97999199}, \sqrt{7.020008}, - \sqrt{7.020008}$ .

$y = \sqrt{56.97999199}, - \sqrt{56.97999199}, \sqrt{7.020008}, - \sqrt{7.020008}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$y = \sqrt{56.97999199}, - \sqrt{56.97999199}, \sqrt{7.020008}, - \sqrt{7.020008}$

Decimal Form:

$y = 7.54850925 \dots, - 7.54850925 \dots, 2.64952977 \dots, - 2.64952977 \dots$