Precalculus Examples

$y^{4} - 4 y^{2} = - 3$

Step 1

Add $3$ to both sides of the equation.

$y^{4} - 4 y^{2} + 3 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

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

${(y^{2})}^{2} - 4 y^{2} + 3 = 0$

Step 2.2

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

$u^{2} - 4 u + 3 = 0$

Step 2.3

Factor $u^{2} - 4 u + 3$ using the AC method.

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Step 2.3.1

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 $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 2.3.2

Write the factored form using these integers.

$(u - 3) (u - 1) = 0$

Step 2.4

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

$(y^{2} - 3) (y^{2} - 1) = 0$

Step 2.5

Rewrite $1$ as $1^{2}$ .

$(y^{2} - 3) (y^{2} - 1^{2}) = 0$

Step 2.6

Factor.

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Step 2.6.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

$(y^{2} - 3) ((y + 1) (y - 1)) = 0$

Step 2.6.2

Remove unnecessary parentheses.

$(y^{2} - 3) (y + 1) (y - 1) = 0$

Step 3

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

$y^{2} - 3 = 0$

$y + 1 = 0$

$y - 1 = 0$

Step 4

Set $y^{2} - 3$ equal to $0$ and solve for $y$ .

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Step 4.1

Set $y^{2} - 3$ equal to $0$ .

$y^{2} - 3 = 0$

Step 4.2

Solve $y^{2} - 3 = 0$ for $y$ .

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Step 4.2.1

Add $3$ to both sides of the equation.

$y^{2} = 3$

Step 4.2.2

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

$y = \pm \sqrt{3}$

Step 4.2.3

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

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Step 4.2.3.1

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

$y = \sqrt{3}$

Step 4.2.3.2

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

$y = - \sqrt{3}$

Step 4.2.3.3

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

$y = \sqrt{3}, - \sqrt{3}$

Step 5

Set $y + 1$ equal to $0$ and solve for $y$ .

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Step 5.1

Set $y + 1$ equal to $0$ .

$y + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$y = - 1$

Step 6

Set $y - 1$ equal to $0$ and solve for $y$ .

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Step 6.1

Set $y - 1$ equal to $0$ .

$y - 1 = 0$

Step 6.2

Add $1$ to both sides of the equation.

$y = 1$

Step 7

The final solution is all the values that make $(y^{2} - 3) (y + 1) (y - 1) = 0$ true.

$y = \sqrt{3}, - \sqrt{3}, - 1, 1$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = \sqrt{3}, - \sqrt{3}, - 1, 1$

Decimal Form:

$y = 1.73205080 \dots, - 1.73205080 \dots, - 1, 1$