Solve for t 8t = square root of 64t^2+9t-19

$8 t = \sqrt{64 t^{2} + 9 t - 19}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{64 t^{2} + 9 t - 19} = 8 t$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{64 t^{2} + 9 t - 19}}^{2} = {(8 t)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{64 t^{2} + 9 t - 19}$ as ${(64 t^{2} + 9 t - 19)}^{\frac{1}{2}}$ .

${({(64 t^{2} + 9 t - 19)}^{\frac{1}{2}})}^{2} = {(8 t)}^{2}$

Simplify the left side.

Simplify ${({(64 t^{2} + 9 t - 19)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(64 t^{2} + 9 t - 19)}^{\frac{1}{2}})}^{2}$ .

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

${(64 t^{2} + 9 t - 19)}^{\frac{1}{2} \cdot 2} = {(8 t)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(64 t^{2} + 9 t - 19)}^{\frac{1}{2} \cdot 2} = {(8 t)}^{2}$

Rewrite the expression.

${(64 t^{2} + 9 t - 19)}^{1} = {(8 t)}^{2}$

Simplify.

$64 t^{2} + 9 t - 19 = {(8 t)}^{2}$

Simplify the right side.

Simplify ${(8 t)}^{2}$ .

Apply the product rule to $8 t$ .

$64 t^{2} + 9 t - 19 = 8^{2} t^{2}$

Raise $8$ to the power of $2$ .

$64 t^{2} + 9 t - 19 = 64 t^{2}$

Step 4

Solve for $t$ .

Move all terms containing $t$ to the left side of the equation.

Subtract $64 t^{2}$ from both sides of the equation.

$64 t^{2} + 9 t - 19 - 64 t^{2} = 0$

Combine the opposite terms in $64 t^{2} + 9 t - 19 - 64 t^{2}$ .

Subtract $64 t^{2}$ from $64 t^{2}$ .

$9 t - 19 + 0 = 0$

Add $9 t - 19$ and $0$ .

$9 t - 19 = 0$

Add $19$ to both sides of the equation.

$9 t = 19$

Divide each term in $9 t = 19$ by $9$ and simplify.

Divide each term in $9 t = 19$ by $9$ .

$\frac{9 t}{9} = \frac{19}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 t}{9} = \frac{19}{9}$

Divide $t$ by $1$ .

$t = \frac{19}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$t = \frac{19}{9}$

Decimal Form:

$t = 2.\overline{1}$

Mixed Number Form:

$t = 2 \frac{1}{9}$

Solve for t 8t = square root of 64t^2+9t-19

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