Solve for y y^4-7y^2=-10

$y^{4} - 7 y^{2} = - 10$

Step 1

Add $10$ to both sides of the equation.

$y^{4} - 7 y^{2} + 10 = 0$

Step 2

Factor the left side of the equation.

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

${(y^{2})}^{2} - 7 y^{2} + 10 = 0$

Let $u = y^{2}$ . Substitute $u$ for all occurrences of $y^{2}$ .

$u^{2} - 7 u + 10 = 0$

Factor $u^{2} - 7 u + 10$ 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 $10$ and whose sum is $- 7$ .

$- 5, - 2$

Write the factored form using these integers.

$(u - 5) (u - 2) = 0$

Replace all occurrences of $u$ with $y^{2}$ .

$(y^{2} - 5) (y^{2} - 2) = 0$

Step 3

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

$y^{2} - 5 = 0$

$y^{2} - 2 = 0$

Step 4

Set $y^{2} - 5$ equal to $0$ and solve for $y$ .

Set $y^{2} - 5$ equal to $0$ .

$y^{2} - 5 = 0$

Solve $y^{2} - 5 = 0$ for $y$ .

Add $5$ to both sides of the equation.

$y^{2} = 5$

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

$y = \pm \sqrt{5}$

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.

$y = \sqrt{5}$

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

$y = - \sqrt{5}$

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

$y = \sqrt{5}, - \sqrt{5}$

Step 5

Set $y^{2} - 2$ equal to $0$ and solve for $y$ .

Set $y^{2} - 2$ equal to $0$ .

$y^{2} - 2 = 0$

Solve $y^{2} - 2 = 0$ for $y$ .

Add $2$ to both sides of the equation.

$y^{2} = 2$

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

$y = \pm \sqrt{2}$

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.

$y = \sqrt{2}$

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

$y = - \sqrt{2}$

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

$y = \sqrt{2}, - \sqrt{2}$

Step 6

The final solution is all the values that make $(y^{2} - 5) (y^{2} - 2) = 0$ true.

$y = \sqrt{5}, - \sqrt{5}, \sqrt{2}, - \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = \sqrt{5}, - \sqrt{5}, \sqrt{2}, - \sqrt{2}$

Decimal Form:

$y = 2.23606797 \dots, - 2.23606797 \dots, 1.41421356 \dots, - 1.41421356 \dots$

Solve for y y^4-7y^2=-10

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