Solve Using the Quadratic Formula 3(t^2-9)^2+16(t^2-9)=-5

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

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

Simplify the left side.

Simplify each term.

Apply the distributive property.

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

Multiply $16$ by $- 9$ .

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

Add $5$ to both sides of the equation.

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

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

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

Add $2$ and $2$ .

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

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

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

Multiply $- 9$ by $- 9$ .

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

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

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

Apply the distributive property.

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

Simplify.

Multiply $- 18$ by $3$ .

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

Multiply $3$ by $81$ .

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

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

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

Subtract $144$ from $243$ .

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

Add $99$ and $5$ .

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

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

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

$u = t^{2}$

Factor by grouping.

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 $b = - 38$ .

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

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

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

$u - 4 = 0$

$3 u - 26 = 0$

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

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

$u - 4 = 0$

Add $4$ to both sides of the equation.

$u = 4$

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

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

$3 u - 26 = 0$

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

Add $26$ to both sides of the equation.

$3 u = 26$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $u$ by $1$ .

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

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

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

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

$t^{2} = 4$

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

Solve the first equation for $t$ .

$t^{2} = 4$

Solve the equation for $t$ .

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

$t = \pm \sqrt{4}$

Simplify $\pm \sqrt{4}$ .

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

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

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

$t = \pm 2$

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.

$t = 2$

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

$t = - 2$

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

$t = 2, - 2$

Solve the second equation for $t$ .

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

Solve the equation for $t$ .

Remove parentheses.

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

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

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

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

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $26$ by $3$ .

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

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.

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

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

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

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

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

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

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

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

Solve Using the Quadratic Formula 3(t^2-9)^2+16(t^2-9)=-5

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