Precalculus Examples

$6 w^{4} - 17 w^{2} + 12 = 0$

Step 1

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

$6 u^{2} - 17 u + 12 = 0$

$u = w^{2}$

Step 2

Factor by grouping.

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Step 2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 6 \cdot 12 = 72$ and whose sum is $b = - 17$ .

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Step 2.1.1

Factor $- 17$ out of $- 17 u$ .

$6 u^{2} - 17 u + 12 = 0$

Step 2.1.2

Rewrite $- 17$ as $- 8$ plus $- 9$

$6 u^{2} + (- 8 - 9) u + 12 = 0$

Step 2.1.3

Apply the distributive property.

$6 u^{2} - 8 u - 9 u + 12 = 0$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$(6 u^{2} - 8 u) - 9 u + 12 = 0$

Step 2.2.2

Factor out the greatest common factor (GCF) from each group.

$2 u (3 u - 4) - 3 (3 u - 4) = 0$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $3 u - 4$ .

$(3 u - 4) (2 u - 3) = 0$

Step 3

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

$3 u - 4 = 0$

$2 u - 3 = 0$

Step 4

Set $3 u - 4$ equal to $0$ and solve for $u$ .

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Step 4.1

Set $3 u - 4$ equal to $0$ .

$3 u - 4 = 0$

Step 4.2

Solve $3 u - 4 = 0$ for $u$ .

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Step 4.2.1

Add $4$ to both sides of the equation.

$3 u = 4$

Step 4.2.2

Divide each term in $3 u = 4$ by $3$ and simplify.

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Step 4.2.2.1

Divide each term in $3 u = 4$ by $3$ .

$\frac{3 u}{3} = \frac{4}{3}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{3 u}{3} = \frac{4}{3}$

Step 4.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{4}{3}$

Step 5

Set $2 u - 3$ equal to $0$ and solve for $u$ .

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Step 5.1

Set $2 u - 3$ equal to $0$ .

$2 u - 3 = 0$

Step 5.2

Solve $2 u - 3 = 0$ for $u$ .

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Step 5.2.1

Add $3$ to both sides of the equation.

$2 u = 3$

Step 5.2.2

Divide each term in $2 u = 3$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 u = 3$ by $2$ .

$\frac{2 u}{2} = \frac{3}{2}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{3}{2}$

Step 5.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{3}{2}$

Step 6

The final solution is all the values that make $(3 u - 4) (2 u - 3) = 0$ true.

$u = \frac{4}{3}, \frac{3}{2}$

Step 7

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

$w^{2} = \frac{4}{3}$

${(w^{2})}^{1} = \frac{3}{2}$

Step 8

Solve the first equation for $w$ .

$w^{2} = \frac{4}{3}$

Step 9

Solve the equation for $w$ .

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Step 9.1

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

$w = \pm \sqrt{\frac{4}{3}}$

Step 9.2

Simplify $\pm \sqrt{\frac{4}{3}}$ .

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Step 9.2.1

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$w = \pm \frac{\sqrt{4}}{\sqrt{3}}$

Step 9.2.2

Simplify the numerator.

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Step 9.2.2.1

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

$w = \pm \frac{\sqrt{2^{2}}}{\sqrt{3}}$

Step 9.2.2.2

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

$w = \pm \frac{2}{\sqrt{3}}$

Step 9.2.3

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$w = \pm \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 9.2.4

Combine and simplify the denominator.

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Step 9.2.4.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$w = \pm \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 9.2.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$w = \pm \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 9.2.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$w = \pm \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 9.2.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$w = \pm \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 9.2.4.5

Add $1$ and $1$ .

$w = \pm \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 9.2.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 9.2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$w = \pm \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 9.2.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$w = \pm \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 9.2.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$w = \pm \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 9.2.4.6.4

Cancel the common factor of $2$ .

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Step 9.2.4.6.4.1

Cancel the common factor.

$w = \pm \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 9.2.4.6.4.2

Rewrite the expression.

$w = \pm \frac{2 \sqrt{3}}{3^{1}}$

Step 9.2.4.6.5

Evaluate the exponent.

$w = \pm \frac{2 \sqrt{3}}{3}$

Step 9.3

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

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Step 9.3.1

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

$w = \frac{2 \sqrt{3}}{3}$

Step 9.3.2

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

$w = - \frac{2 \sqrt{3}}{3}$

Step 9.3.3

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

$w = \frac{2 \sqrt{3}}{3}, - \frac{2 \sqrt{3}}{3}$

Step 10

Solve the second equation for $w$ .

${(w^{2})}^{1} = \frac{3}{2}$

Step 11

Solve the equation for $w$ .

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Step 11.1

Remove parentheses.

$w^{2} = \frac{3}{2}$

Step 11.2

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

$w = \pm \sqrt{\frac{3}{2}}$

Step 11.3

Simplify $\pm \sqrt{\frac{3}{2}}$ .

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Step 11.3.1

Rewrite $\sqrt{\frac{3}{2}}$ as $\frac{\sqrt{3}}{\sqrt{2}}$ .

$w = \pm \frac{\sqrt{3}}{\sqrt{2}}$

Step 11.3.2

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$w = \pm \frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 11.3.3

Combine and simplify the denominator.

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Step 11.3.3.1

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$w = \pm \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 11.3.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$w = \pm \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 11.3.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$w = \pm \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 11.3.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$w = \pm \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 11.3.3.5

Add $1$ and $1$ .

$w = \pm \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 11.3.3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 11.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$w = \pm \frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 11.3.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$w = \pm \frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 11.3.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$w = \pm \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 11.3.3.6.4

Cancel the common factor of $2$ .

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Step 11.3.3.6.4.1

Cancel the common factor.

$w = \pm \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 11.3.3.6.4.2

Rewrite the expression.

$w = \pm \frac{\sqrt{3} \sqrt{2}}{2^{1}}$

Step 11.3.3.6.5

Evaluate the exponent.

$w = \pm \frac{\sqrt{3} \sqrt{2}}{2}$

Step 11.3.4

Simplify the numerator.

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Step 11.3.4.1

Combine using the product rule for radicals.

$w = \pm \frac{\sqrt{3 \cdot 2}}{2}$

Step 11.3.4.2

Multiply $3$ by $2$ .

$w = \pm \frac{\sqrt{6}}{2}$

Step 11.4

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

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Step 11.4.1

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

$w = \frac{\sqrt{6}}{2}$

Step 11.4.2

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

$w = - \frac{\sqrt{6}}{2}$

Step 11.4.3

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

$w = \frac{\sqrt{6}}{2}, - \frac{\sqrt{6}}{2}$

Step 12

The solution to $6 w^{4} - 17 w^{2} + 12 = 0$ is $w = \frac{2 \sqrt{3}}{3}, - \frac{2 \sqrt{3}}{3}, \frac{\sqrt{6}}{2}, - \frac{\sqrt{6}}{2}$ .

$w = \frac{2 \sqrt{3}}{3}, - \frac{2 \sqrt{3}}{3}, \frac{\sqrt{6}}{2}, - \frac{\sqrt{6}}{2}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$w = \frac{2 \sqrt{3}}{3}, - \frac{2 \sqrt{3}}{3}, \frac{\sqrt{6}}{2}, - \frac{\sqrt{6}}{2}$

Decimal Form:

$w = 1.15470053 \dots, - 1.15470053 \dots, 1.22474487 \dots, - 1.22474487 \dots$