Solve 9-7 5 72 9t - 7 = 3t - 4 | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$5 \cdot \frac { 9 t - 7 } { 3 t + 5 } = \frac { 3 t - 4 } { t - 6 }$$

Solve for t

$t = \frac{\sqrt{3859} + 77}{18} \approx 7.728936109$
$t=\frac{77-\sqrt{3859}}{18}\approx 0.826619447$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Variable $t$ cannot be equal to any of the values $-\frac{5}{3},6$ since division by zero is not defined. Multiply both sides of the equation by $\left(t-6\right)\left(3t+5\right)$, the least common multiple of $3t+5,t-6$.

$$5\left(t-6\right)\left(9t-7\right)=\left(3t+5\right)\left(3t-4\right)$$

Use the distributive property to multiply $5$ by $t-6$.

$$\left(5t-30\right)\left(9t-7\right)=\left(3t+5\right)\left(3t-4\right)$$

Use the distributive property to multiply $5t-30$ by $9t-7$ and combine like terms.

$$45t^{2}-305t+210=\left(3t+5\right)\left(3t-4\right)$$

Use the distributive property to multiply $3t+5$ by $3t-4$ and combine like terms.

$$45t^{2}-305t+210=9t^{2}+3t-20$$

Subtract $9t^{2}$ from both sides.

$$45t^{2}-305t+210-9t^{2}=3t-20$$

Combine $45t^{2}$ and $-9t^{2}$ to get $36t^{2}$.

$$36t^{2}-305t+210=3t-20$$

Subtract $3t$ from both sides.

$$36t^{2}-305t+210-3t=-20$$

Combine $-305t$ and $-3t$ to get $-308t$.

$$36t^{2}-308t+210=-20$$

Add $20$ to both sides.

$$36t^{2}-308t+210+20=0$$

Add $210$ and $20$ to get $230$.

$$36t^{2}-308t+230=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $36$ for $a$, $-308$ for $b$, and $230$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$t=\frac{-\left(-308\right)±\sqrt{\left(-308\right)^{2}-4\times 36\times 230}}{2\times 36}$$

Square $-308$.

$$t=\frac{-\left(-308\right)±\sqrt{94864-4\times 36\times 230}}{2\times 36}$$

Multiply $-4$ times $36$.

$$t=\frac{-\left(-308\right)±\sqrt{94864-144\times 230}}{2\times 36}$$

Multiply $-144$ times $230$.

$$t=\frac{-\left(-308\right)±\sqrt{94864-33120}}{2\times 36}$$

Add $94864$ to $-33120$.

$$t=\frac{-\left(-308\right)±\sqrt{61744}}{2\times 36}$$

Take the square root of $61744$.

$$t=\frac{-\left(-308\right)±4\sqrt{3859}}{2\times 36}$$

The opposite of $-308$ is $308$.

$$t=\frac{308±4\sqrt{3859}}{2\times 36}$$

Multiply $2$ times $36$.

$$t=\frac{308±4\sqrt{3859}}{72}$$

Now solve the equation $t=\frac{308±4\sqrt{3859}}{72}$ when $±$ is plus. Add $308$ to $4\sqrt{3859}$.

$$t=\frac{4\sqrt{3859}+308}{72}$$

Divide $308+4\sqrt{3859}$ by $72$.

$$t=\frac{\sqrt{3859}+77}{18}$$

Now solve the equation $t=\frac{308±4\sqrt{3859}}{72}$ when $±$ is minus. Subtract $4\sqrt{3859}$ from $308$.

$$t=\frac{308-4\sqrt{3859}}{72}$$

Divide $308-4\sqrt{3859}$ by $72$.

$$t=\frac{77-\sqrt{3859}}{18}$$

The equation is now solved.

$$t=\frac{\sqrt{3859}+77}{18}$$ $$t=\frac{77-\sqrt{3859}}{18}$$

Steps for Completing the Square

Variable $t$ cannot be equal to any of the values $-\frac{5}{3},6$ since division by zero is not defined. Multiply both sides of the equation by $\left(t-6\right)\left(3t+5\right)$, the least common multiple of $3t+5,t-6$.

$$5\left(t-6\right)\left(9t-7\right)=\left(3t+5\right)\left(3t-4\right)$$

Use the distributive property to multiply $5$ by $t-6$.

$$\left(5t-30\right)\left(9t-7\right)=\left(3t+5\right)\left(3t-4\right)$$

Use the distributive property to multiply $5t-30$ by $9t-7$ and combine like terms.

$$45t^{2}-305t+210=\left(3t+5\right)\left(3t-4\right)$$

Use the distributive property to multiply $3t+5$ by $3t-4$ and combine like terms.

$$45t^{2}-305t+210=9t^{2}+3t-20$$

Subtract $9t^{2}$ from both sides.

$$45t^{2}-305t+210-9t^{2}=3t-20$$

Combine $45t^{2}$ and $-9t^{2}$ to get $36t^{2}$.

$$36t^{2}-305t+210=3t-20$$

Subtract $3t$ from both sides.

$$36t^{2}-305t+210-3t=-20$$

Combine $-305t$ and $-3t$ to get $-308t$.

$$36t^{2}-308t+210=-20$$

Subtract $210$ from both sides.

$$36t^{2}-308t=-20-210$$

Subtract $210$ from $-20$ to get $-230$.

$$36t^{2}-308t=-230$$

Divide both sides by $36$.

$$\frac{36t^{2}-308t}{36}=-\frac{230}{36}$$

Dividing by $36$ undoes the multiplication by $36$.

$$t^{2}+\left(-\frac{308}{36}\right)t=-\frac{230}{36}$$

Reduce the fraction $\frac{-308}{36}$ to lowest terms by extracting and canceling out $4$.

$$t^{2}-\frac{77}{9}t=-\frac{230}{36}$$

Reduce the fraction $\frac{-230}{36}$ to lowest terms by extracting and canceling out $2$.

$$t^{2}-\frac{77}{9}t=-\frac{115}{18}$$

Divide $-\frac{77}{9}$, the coefficient of the $x$ term, by $2$ to get $-\frac{77}{18}$. Then add the square of $-\frac{77}{18}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{77}{9}t+\left(-\frac{77}{18}\right)^{2}=-\frac{115}{18}+\left(-\frac{77}{18}\right)^{2}$$

Square $-\frac{77}{18}$ by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{77}{9}t+\frac{5929}{324}=-\frac{115}{18}+\frac{5929}{324}$$

Add $-\frac{115}{18}$ to $\frac{5929}{324}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{77}{9}t+\frac{5929}{324}=\frac{3859}{324}$$

Factor $t^{2}-\frac{77}{9}t+\frac{5929}{324}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(t-\frac{77}{18}\right)^{2}=\frac{3859}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{77}{18}\right)^{2}}=\sqrt{\frac{3859}{324}}$$

Simplify.

$$t-\frac{77}{18}=\frac{\sqrt{3859}}{18}$$ $$t-\frac{77}{18}=-\frac{\sqrt{3859}}{18}$$

Add $\frac{77}{18}$ to both sides of the equation.

$$t=\frac{\sqrt{3859}+77}{18}$$ $$t=\frac{77-\sqrt{3859}}{18}$$