Evaluate 3(t^2-16)^2+19(t^2-16)=-6

$3 {(t^{2} - 16)}^{2} + 19 (t^{2} - 16) = - 6$

Simplify each term.

Apply the distributive property.

$3 {(t^{2} - 16)}^{2} + 19 t^{2} + 19 \cdot - 16 = - 6$

Multiply $19$ by $- 16$ .

$3 {(t^{2} - 16)}^{2} + 19 t^{2} - 304 = - 6$

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

$3 u^{2} - 77 u + 464 = - 6$

$u = t^{2}$

Add $6$ to both sides of the equation.

$3 u^{2} - 77 u + 464 + 6 = 0$

Add $464$ and $6$ .

$3 u^{2} - 77 u + 470 = 0$

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 $a \cdot c = 3 \cdot 470 = 1410$ and whose sum is $b = - 77$ .

Factor $- 77$ out of $- 77 u$ .

$3 u^{2} - 77 u + 470 = 0$

Rewrite $- 77$ as $- 30$ plus $- 47$

$3 u^{2} + (- 30 - 47) u + 470 = 0$

Apply the distributive property.

$3 u^{2} - 30 u - 47 u + 470 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 u^{2} - 30 u) - 47 u + 470 = 0$

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

$3 u (u - 10) - 47 (u - 10) = 0$

Factor the polynomial by factoring out the greatest common factor, $u - 10$ .

$(u - 10) (3 u - 47) = 0$

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

$u - 10 = 0$

$3 u - 47 = 0$

Set $u - 10$ equal to $0$ and solve for $u$ .

Set $u - 10$ equal to $0$ .

$u - 10 = 0$

Add $10$ to both sides of the equation.

$u = 10$

Set $3 u - 47$ equal to $0$ and solve for $u$ .

Set $3 u - 47$ equal to $0$ .

$3 u - 47 = 0$

Solve $3 u - 47 = 0$ for $u$ .

Add $47$ to both sides of the equation.

$3 u = 47$

Divide each term in $3 u = 47$ by $3$ and simplify.

Divide each term in $3 u = 47$ by $3$ .

$\frac{3 u}{3} = \frac{47}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 u}{3} = \frac{47}{3}$

Divide $u$ by $1$ .

$u = \frac{47}{3}$

The final solution is all the values that make $(u - 10) (3 u - 47) = 0$ true.

$u = 10, \frac{47}{3}$

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

$t^{2} = 10$

${(t^{2})}^{1} = \frac{47}{3}$

Solve the first equation for $t$ .

$t^{2} = 10$

Solve the equation for $t$ .

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

$t = \pm \sqrt{10}$

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.

$t = \sqrt{10}$

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

$t = - \sqrt{10}$

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

$t = \sqrt{10}, - \sqrt{10}$

Solve the second equation for $t$ .

${(t^{2})}^{1} = \frac{47}{3}$

Solve the equation for $t$ .

Remove parentheses.

$t^{2} = \frac{47}{3}$

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

$t = \pm \sqrt{\frac{47}{3}}$

Simplify $\pm \sqrt{\frac{47}{3}}$ .

Rewrite $\sqrt{\frac{47}{3}}$ as $\frac{\sqrt{47}}{\sqrt{3}}$ .

$t = \pm \frac{\sqrt{47}}{\sqrt{3}}$

Multiply $\frac{\sqrt{47}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$t = \pm \frac{\sqrt{47}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{47}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$t = \pm \frac{\sqrt{47} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$t = \pm \frac{\sqrt{47} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

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

$t = \pm \frac{\sqrt{47} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

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

$t = \pm \frac{\sqrt{47} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$t = \pm \frac{\sqrt{47} \sqrt{3}}{{\sqrt{3}}^{2}}$

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

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

$t = \pm \frac{\sqrt{47} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$t = \pm \frac{\sqrt{47} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

$t = \pm \frac{\sqrt{47} \sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$t = \pm \frac{\sqrt{47} \sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$t = \pm \frac{\sqrt{47} \sqrt{3}}{3^{1}}$

Evaluate the exponent.

$t = \pm \frac{\sqrt{47} \sqrt{3}}{3}$

Simplify the numerator.

Combine using the product rule for radicals.

$t = \pm \frac{\sqrt{47 \cdot 3}}{3}$

Multiply $47$ by $3$ .

$t = \pm \frac{\sqrt{141}}{3}$

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.

$t = \frac{\sqrt{141}}{3}$

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

$t = - \frac{\sqrt{141}}{3}$

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

$t = \frac{\sqrt{141}}{3}, - \frac{\sqrt{141}}{3}$

The solution to $3 t^{4} - 77 t^{2} + 464 = - 6$ is $t = \sqrt{10}, - \sqrt{10}, \frac{\sqrt{141}}{3}, - \frac{\sqrt{141}}{3}$ .

$t = \sqrt{10}, - \sqrt{10}, \frac{\sqrt{141}}{3}, - \frac{\sqrt{141}}{3}$

The result can be shown in multiple forms.

Exact Form:

$t = \sqrt{10}, - \sqrt{10}, \frac{\sqrt{141}}{3}, - \frac{\sqrt{141}}{3}$

Decimal Form:

$t = 3.16227766 \dots, - 3.16227766 \dots, 3.95811402 \dots, - 3.95811402 \dots$

Evaluate 3(t^2-16)^2+19(t^2-16)=-6

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