Solve (9)/(3)x+(4)/(5)x=(16x)/(2)+(13x)/(4)x | 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

$$\frac{ 9 }{ 3 } x+ \frac{ 4 }{ 5 } x= \frac{ 16x }{ 2 } + \frac{ 13x }{ 4 } x$$

Solve for x

$x = -\frac{84}{65} = -1\frac{19}{65} \approx -1.292307692$
$x=0$

Steps Using Factoring

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using Factoring

Multiply both sides of the equation by $60$, the least common multiple of $3,5,2,4$.

$$180x+48x=30\times 16x+15\times 13xx$$

Combine $180x$ and $48x$ to get $228x$.

$$228x=30\times 16x+15\times 13xx$$

Multiply $x$ and $x$ to get $x^{2}$.

$$228x=30\times 16x+15\times 13x^{2}$$

Multiply $30$ and $16$ to get $480$. Multiply $15$ and $13$ to get $195$.

$$228x=480x+195x^{2}$$

Subtract $480x$ from both sides.

$$228x-480x=195x^{2}$$

Combine $228x$ and $-480x$ to get $-252x$.

$$-252x=195x^{2}$$

Subtract $195x^{2}$ from both sides.

$$-252x-195x^{2}=0$$

Factor out $x$.

$$x\left(-252-195x\right)=0$$

To find equation solutions, solve $x=0$ and $-252-195x=0$.

$$x=0$$ $$x=-\frac{84}{65}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by $60$, the least common multiple of $3,5,2,4$.

$$180x+48x=30\times 16x+15\times 13xx$$

Combine $180x$ and $48x$ to get $228x$.

$$228x=30\times 16x+15\times 13xx$$

Multiply $x$ and $x$ to get $x^{2}$.

$$228x=30\times 16x+15\times 13x^{2}$$

Multiply $30$ and $16$ to get $480$. Multiply $15$ and $13$ to get $195$.

$$228x=480x+195x^{2}$$

Subtract $480x$ from both sides.

$$228x-480x=195x^{2}$$

Combine $228x$ and $-480x$ to get $-252x$.

$$-252x=195x^{2}$$

Subtract $195x^{2}$ from both sides.

$$-252x-195x^{2}=0$$

All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.

$$-195x^{2}-252x=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-195$ for $a$, $-252$ for $b$, and $0$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-\left(-252\right)±\sqrt{\left(-252\right)^{2}}}{2\left(-195\right)}$$

Take the square root of $\left(-252\right)^{2}$.

$$x=\frac{-\left(-252\right)±252}{2\left(-195\right)}$$

The opposite of $-252$ is $252$.

$$x=\frac{252±252}{2\left(-195\right)}$$

Multiply $2$ times $-195$.

$$x=\frac{252±252}{-390}$$

Now solve the equation $x=\frac{252±252}{-390}$ when $±$ is plus. Add $252$ to $252$.

$$x=\frac{504}{-390}$$

Reduce the fraction $\frac{504}{-390}$ to lowest terms by extracting and canceling out $6$.

$$x=-\frac{84}{65}$$

Now solve the equation $x=\frac{252±252}{-390}$ when $±$ is minus. Subtract $252$ from $252$.

$$x=\frac{0}{-390}$$

Divide $0$ by $-390$.

$$x=0$$

The equation is now solved.

$$x=-\frac{84}{65}$$ $$x=0$$

Steps for Completing the Square

Multiply both sides of the equation by $60$, the least common multiple of $3,5,2,4$.

$$180x+48x=30\times 16x+15\times 13xx$$

Combine $180x$ and $48x$ to get $228x$.

$$228x=30\times 16x+15\times 13xx$$

Multiply $x$ and $x$ to get $x^{2}$.

$$228x=30\times 16x+15\times 13x^{2}$$

Multiply $30$ and $16$ to get $480$. Multiply $15$ and $13$ to get $195$.

$$228x=480x+195x^{2}$$

Subtract $480x$ from both sides.

$$228x-480x=195x^{2}$$

Combine $228x$ and $-480x$ to get $-252x$.

$$-252x=195x^{2}$$

Subtract $195x^{2}$ from both sides.

$$-252x-195x^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form $x^{2}+bx=c$.

$$-195x^{2}-252x=0$$

Divide both sides by $-195$.

$$\frac{-195x^{2}-252x}{-195}=\frac{0}{-195}$$

Dividing by $-195$ undoes the multiplication by $-195$.

$$x^{2}+\left(-\frac{252}{-195}\right)x=\frac{0}{-195}$$

Reduce the fraction $\frac{-252}{-195}$ to lowest terms by extracting and canceling out $3$.

$$x^{2}+\frac{84}{65}x=\frac{0}{-195}$$

Divide $0$ by $-195$.

$$x^{2}+\frac{84}{65}x=0$$

Divide $\frac{84}{65}$, the coefficient of the $x$ term, by $2$ to get $\frac{42}{65}$. Then add the square of $\frac{42}{65}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{84}{65}x+\left(\frac{42}{65}\right)^{2}=\left(\frac{42}{65}\right)^{2}$$

Square $\frac{42}{65}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{84}{65}x+\frac{1764}{4225}=\frac{1764}{4225}$$

Factor $x^{2}+\frac{84}{65}x+\frac{1764}{4225}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x+\frac{42}{65}\right)^{2}=\frac{1764}{4225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{42}{65}\right)^{2}}=\sqrt{\frac{1764}{4225}}$$

Simplify.

$$x+\frac{42}{65}=\frac{42}{65}$$ $$x+\frac{42}{65}=-\frac{42}{65}$$

Subtract $\frac{42}{65}$ from both sides of the equation.

$$x=0$$ $$x=-\frac{84}{65}$$