Solve 8x - 6 - (2 ^ 7)/(3x) = 0 | 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

$$8x-6-\frac{2^{7}}{3x}=0$$

Solve for x

$x=\frac{\sqrt{3153}}{24}+\frac{3}{8}\approx 2.71464919$
$x=-\frac{\sqrt{3153}}{24}+\frac{3}{8}\approx -1.96464919$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Variable $x$ cannot be equal to $0$ since division by zero is not defined. Multiply both sides of the equation by $3x$.

$$8x\times 3x+3x\left(-6\right)-2^{7}=0$$

Multiply $8$ and $3$ to get $24$.

$$24xx+3x\left(-6\right)-2^{7}=0$$

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

$$24x^{2}+3x\left(-6\right)-2^{7}=0$$

Multiply $3$ and $-6$ to get $-18$.

$$24x^{2}-18x-2^{7}=0$$

Calculate $2$ to the power of $7$ and get $128$.

$$24x^{2}-18x-128=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $24$ for $a$, $-18$ for $b$, and $-128$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 24\left(-128\right)}}{2\times 24}$$

Square $-18$.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 24\left(-128\right)}}{2\times 24}$$

Multiply $-4$ times $24$.

$$x=\frac{-\left(-18\right)±\sqrt{324-96\left(-128\right)}}{2\times 24}$$

Multiply $-96$ times $-128$.

$$x=\frac{-\left(-18\right)±\sqrt{324+12288}}{2\times 24}$$

Add $324$ to $12288$.

$$x=\frac{-\left(-18\right)±\sqrt{12612}}{2\times 24}$$

Take the square root of $12612$.

$$x=\frac{-\left(-18\right)±2\sqrt{3153}}{2\times 24}$$

The opposite of $-18$ is $18$.

$$x=\frac{18±2\sqrt{3153}}{2\times 24}$$

Multiply $2$ times $24$.

$$x=\frac{18±2\sqrt{3153}}{48}$$

Now solve the equation $x=\frac{18±2\sqrt{3153}}{48}$ when $±$ is plus. Add $18$ to $2\sqrt{3153}$.

$$x=\frac{2\sqrt{3153}+18}{48}$$

Divide $18+2\sqrt{3153}$ by $48$.

$$x=\frac{\sqrt{3153}}{24}+\frac{3}{8}$$

Now solve the equation $x=\frac{18±2\sqrt{3153}}{48}$ when $±$ is minus. Subtract $2\sqrt{3153}$ from $18$.

$$x=\frac{18-2\sqrt{3153}}{48}$$

Divide $18-2\sqrt{3153}$ by $48$.

$$x=-\frac{\sqrt{3153}}{24}+\frac{3}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{3153}}{24}+\frac{3}{8}$$ $$x=-\frac{\sqrt{3153}}{24}+\frac{3}{8}$$

Steps for Completing the Square

Variable $x$ cannot be equal to $0$ since division by zero is not defined. Multiply both sides of the equation by $3x$.

$$8x\times 3x+3x\left(-6\right)-2^{7}=0$$

Multiply $8$ and $3$ to get $24$.

$$24xx+3x\left(-6\right)-2^{7}=0$$

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

$$24x^{2}+3x\left(-6\right)-2^{7}=0$$

Multiply $3$ and $-6$ to get $-18$.

$$24x^{2}-18x-2^{7}=0$$

Calculate $2$ to the power of $7$ and get $128$.

$$24x^{2}-18x-128=0$$

Add $128$ to both sides. Anything plus zero gives itself.

$$24x^{2}-18x=128$$

Divide both sides by $24$.

$$\frac{24x^{2}-18x}{24}=\frac{128}{24}$$

Dividing by $24$ undoes the multiplication by $24$.

$$x^{2}+\left(-\frac{18}{24}\right)x=\frac{128}{24}$$

Reduce the fraction $\frac{-18}{24}$ to lowest terms by extracting and canceling out $6$.

$$x^{2}-\frac{3}{4}x=\frac{128}{24}$$

Reduce the fraction $\frac{128}{24}$ to lowest terms by extracting and canceling out $8$.

$$x^{2}-\frac{3}{4}x=\frac{16}{3}$$

Divide $-\frac{3}{4}$, the coefficient of the $x$ term, by $2$ to get $-\frac{3}{8}$. Then add the square of $-\frac{3}{8}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{4}x+\left(-\frac{3}{8}\right)^{2}=\frac{16}{3}+\left(-\frac{3}{8}\right)^{2}$$

Square $-\frac{3}{8}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=\frac{16}{3}+\frac{9}{64}$$

Add $\frac{16}{3}$ to $\frac{9}{64}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=\frac{1051}{192}$$

Factor $x^{2}-\frac{3}{4}x+\frac{9}{64}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-\frac{3}{8}\right)^{2}=\frac{1051}{192}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{8}\right)^{2}}=\sqrt{\frac{1051}{192}}$$

Simplify.

$$x-\frac{3}{8}=\frac{\sqrt{3153}}{24}$$ $$x-\frac{3}{8}=-\frac{\sqrt{3153}}{24}$$

Add $\frac{3}{8}$ to both sides of the equation.

$$x=\frac{\sqrt{3153}}{24}+\frac{3}{8}$$ $$x=-\frac{\sqrt{3153}}{24}+\frac{3}{8}$$