Algebra Examples

$3 {(t^{2} - 9)}^{2} + 16 (t^{2} - 9) = - 5$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Apply the distributive property.

$3 {(t^{2} - 9)}^{2} + 16 t^{2} + 16 \cdot - 9 = - 5$

Step 1.1.1.2

Multiply $16$ by $- 9$ .

$3 {(t^{2} - 9)}^{2} + 16 t^{2} - 144 = - 5$

Step 1.2

Add $5$ to both sides of the equation.

$3 {(t^{2} - 9)}^{2} + 16 t^{2} - 144 + 5 = 0$

Step 1.3

Simplify $3 {(t^{2} - 9)}^{2} + 16 t^{2} - 144 + 5$ .

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite ${(t^{2} - 9)}^{2}$ as $(t^{2} - 9) (t^{2} - 9)$ .

$3 ((t^{2} - 9) (t^{2} - 9)) + 16 t^{2} - 144 + 5 = 0$

Step 1.3.1.2

Expand $(t^{2} - 9) (t^{2} - 9)$ using the FOIL Method.

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Step 1.3.1.2.1

Apply the distributive property.

$3 (t^{2} (t^{2} - 9) - 9 (t^{2} - 9)) + 16 t^{2} - 144 + 5 = 0$

Step 1.3.1.2.2

Apply the distributive property.

$3 (t^{2} t^{2} + t^{2} \cdot - 9 - 9 (t^{2} - 9)) + 16 t^{2} - 144 + 5 = 0$

Step 1.3.1.2.3

Apply the distributive property.

$3 (t^{2} t^{2} + t^{2} \cdot - 9 - 9 t^{2} - 9 \cdot - 9) + 16 t^{2} - 144 + 5 = 0$

Step 1.3.1.3

Simplify and combine like terms.

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Step 1.3.1.3.1

Simplify each term.

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Step 1.3.1.3.1.1

Multiply $t^{2}$ by $t^{2}$ by adding the exponents.

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Step 1.3.1.3.1.1.1

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

$3 (t^{2 + 2} + t^{2} \cdot - 9 - 9 t^{2} - 9 \cdot - 9) + 16 t^{2} - 144 + 5 = 0$

Step 1.3.1.3.1.1.2

Add $2$ and $2$ .

$3 (t^{4} + t^{2} \cdot - 9 - 9 t^{2} - 9 \cdot - 9) + 16 t^{2} - 144 + 5 = 0$

Step 1.3.1.3.1.2

Move $- 9$ to the left of $t^{2}$ .

$3 (t^{4} - 9 \cdot t^{2} - 9 t^{2} - 9 \cdot - 9) + 16 t^{2} - 144 + 5 = 0$

Step 1.3.1.3.1.3

Multiply $- 9$ by $- 9$ .

$3 (t^{4} - 9 t^{2} - 9 t^{2} + 81) + 16 t^{2} - 144 + 5 = 0$

Step 1.3.1.3.2

Subtract $9 t^{2}$ from $- 9 t^{2}$ .

$3 (t^{4} - 18 t^{2} + 81) + 16 t^{2} - 144 + 5 = 0$

Step 1.3.1.4

Apply the distributive property.

$3 t^{4} + 3 (- 18 t^{2}) + 3 \cdot 81 + 16 t^{2} - 144 + 5 = 0$

Step 1.3.1.5

Simplify.

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Step 1.3.1.5.1

Multiply $- 18$ by $3$ .

$3 t^{4} - 54 t^{2} + 3 \cdot 81 + 16 t^{2} - 144 + 5 = 0$

Step 1.3.1.5.2

Multiply $3$ by $81$ .

$3 t^{4} - 54 t^{2} + 243 + 16 t^{2} - 144 + 5 = 0$

Step 1.3.2

Add $- 54 t^{2}$ and $16 t^{2}$ .

$3 t^{4} - 38 t^{2} + 243 - 144 + 5 = 0$

Step 1.3.3

Subtract $144$ from $243$ .

$3 t^{4} - 38 t^{2} + 99 + 5 = 0$

Step 1.3.4

Add $99$ and $5$ .

$3 t^{4} - 38 t^{2} + 104 = 0$

Step 2

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

$3 u^{2} - 38 u + 104 = 0$

$u = t^{2}$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 3 \cdot 104 = 312$ and whose sum is $b = - 38$ .

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Step 3.1.1

Factor $- 38$ out of $- 38 u$ .

$3 u^{2} - 38 u + 104 = 0$

Step 3.1.2

Rewrite $- 38$ as $- 12$ plus $- 26$

$3 u^{2} + (- 12 - 26) u + 104 = 0$

Step 3.1.3

Apply the distributive property.

$3 u^{2} - 12 u - 26 u + 104 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(3 u^{2} - 12 u) - 26 u + 104 = 0$

Step 3.2.2

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

$3 u (u - 4) - 26 (u - 4) = 0$

Step 3.3

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

$(u - 4) (3 u - 26) = 0$

Step 4

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

$u - 4 = 0$

$3 u - 26 = 0$

Step 5

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

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Step 5.1

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

$u - 4 = 0$

Step 5.2

Add $4$ to both sides of the equation.

$u = 4$

Step 6

Set $3 u - 26$ equal to $0$ and solve for $u$ .

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Step 6.1

Set $3 u - 26$ equal to $0$ .

$3 u - 26 = 0$

Step 6.2

Solve $3 u - 26 = 0$ for $u$ .

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Step 6.2.1

Add $26$ to both sides of the equation.

$3 u = 26$

Step 6.2.2

Divide each term in $3 u = 26$ by $3$ and simplify.

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Step 6.2.2.1

Divide each term in $3 u = 26$ by $3$ .

$\frac{3 u}{3} = \frac{26}{3}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{3 u}{3} = \frac{26}{3}$

Step 6.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{26}{3}$

Step 7

The final solution is all the values that make $(u - 4) (3 u - 26) = 0$ true.

$u = 4, \frac{26}{3}$

Step 8

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

$t^{2} = 4$

${(t^{2})}^{1} = \frac{26}{3}$

Step 9

Solve the first equation for $t$ .

$t^{2} = 4$

Step 10

Solve the equation for $t$ .

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Step 10.1

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

$t = \pm \sqrt{4}$

Step 10.2

Simplify $\pm \sqrt{4}$ .

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Step 10.2.1

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

$t = \pm \sqrt{2^{2}}$

Step 10.2.2

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

$t = \pm 2$

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.

$t = 2$

Step 10.3.2

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

$t = - 2$

Step 10.3.3

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

$t = 2, - 2$

Step 11

Solve the second equation for $t$ .

${(t^{2})}^{1} = \frac{26}{3}$

Step 12

Solve the equation for $t$ .

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Step 12.1

Remove parentheses.

$t^{2} = \frac{26}{3}$

Step 12.2

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

$t = \pm \sqrt{\frac{26}{3}}$

Step 12.3

Simplify $\pm \sqrt{\frac{26}{3}}$ .

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Step 12.3.1

Rewrite $\sqrt{\frac{26}{3}}$ as $\frac{\sqrt{26}}{\sqrt{3}}$ .

$t = \pm \frac{\sqrt{26}}{\sqrt{3}}$

Step 12.3.2

Multiply $\frac{\sqrt{26}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$t = \pm \frac{\sqrt{26}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 12.3.3

Combine and simplify the denominator.

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Step 12.3.3.1

Multiply $\frac{\sqrt{26}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$t = \pm \frac{\sqrt{26} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 12.3.3.2

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

$t = \pm \frac{\sqrt{26} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 12.3.3.3

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

$t = \pm \frac{\sqrt{26} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 12.3.3.4

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

$t = \pm \frac{\sqrt{26} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 12.3.3.5

Add $1$ and $1$ .

$t = \pm \frac{\sqrt{26} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 12.3.3.6

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

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Step 12.3.3.6.1

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

$t = \pm \frac{\sqrt{26} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 12.3.3.6.2

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

$t = \pm \frac{\sqrt{26} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 12.3.3.6.3

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

$t = \pm \frac{\sqrt{26} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 12.3.3.6.4

Cancel the common factor of $2$ .

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Step 12.3.3.6.4.1

Cancel the common factor.

$t = \pm \frac{\sqrt{26} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 12.3.3.6.4.2

Rewrite the expression.

$t = \pm \frac{\sqrt{26} \sqrt{3}}{3^{1}}$

Step 12.3.3.6.5

Evaluate the exponent.

$t = \pm \frac{\sqrt{26} \sqrt{3}}{3}$

Step 12.3.4

Simplify the numerator.

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Step 12.3.4.1

Combine using the product rule for radicals.

$t = \pm \frac{\sqrt{26 \cdot 3}}{3}$

Step 12.3.4.2

Multiply $26$ by $3$ .

$t = \pm \frac{\sqrt{78}}{3}$

Step 12.4

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

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Step 12.4.1

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

$t = \frac{\sqrt{78}}{3}$

Step 12.4.2

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

$t = - \frac{\sqrt{78}}{3}$

Step 12.4.3

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

$t = \frac{\sqrt{78}}{3}, - \frac{\sqrt{78}}{3}$

Step 13

The solution to $3 t^{4} - 38 t^{2} + 104 = 0$ is $t = 2, - 2, \frac{\sqrt{78}}{3}, - \frac{\sqrt{78}}{3}$ .

$t = 2, - 2, \frac{\sqrt{78}}{3}, - \frac{\sqrt{78}}{3}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$t = 2, - 2, \frac{\sqrt{78}}{3}, - \frac{\sqrt{78}}{3}$

Decimal Form:

$t = 2, - 2, 2.94392028 \dots, - 2.94392028 \dots$