Solve 1/8x+3/2=sqrt{x}

$$\frac{ 1 }{ 8 } x+ \frac{ 3 }{ 2 } = \sqrt{ x }$$

Solve for x

$$x=36$$

$$x=4$$

Solution Steps

Square both sides of the equation.

$$\left(\frac{1}{8}x+\frac{3}{2}\right)^{2}=\left(\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{1}{8}x+\frac{3}{2}\right)^{2}.

$$\frac{1}{64}x^{2}+\frac{3}{8}x+\frac{9}{4}=\left(\sqrt{x}\right)^{2}$$

Calculate \sqrt{x} to the power of 2 and get x.

$$\frac{1}{64}x^{2}+\frac{3}{8}x+\frac{9}{4}=x$$

Subtract x from both sides.

$$\frac{1}{64}x^{2}+\frac{3}{8}x+\frac{9}{4}-x=0$$

Combine \frac{3}{8}x and -x to get -\frac{5}{8}x.

$$\frac{1}{64}x^{2}-\frac{5}{8}x+\frac{9}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{64} for a, -\frac{5}{8} for b, and \frac{9}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{5}{8}\right)±\sqrt{\left(-\frac{5}{8}\right)^{2}-4\times \frac{1}{64}\times \frac{9}{4}}}{2\times \frac{1}{64}}$$

Square -\frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{5}{8}\right)±\sqrt{\frac{25}{64}-4\times \frac{1}{64}\times \frac{9}{4}}}{2\times \frac{1}{64}}$$

Multiply -4 times \frac{1}{64}.

$$x=\frac{-\left(-\frac{5}{8}\right)±\sqrt{\frac{25}{64}-\frac{1}{16}\times \frac{9}{4}}}{2\times \frac{1}{64}}$$

Multiply -\frac{1}{16} times \frac{9}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{5}{8}\right)±\sqrt{\frac{25-9}{64}}}{2\times \frac{1}{64}}$$

Add \frac{25}{64} to -\frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{5}{8}\right)±\sqrt{\frac{1}{4}}}{2\times \frac{1}{64}}$$

Take the square root of \frac{1}{4}.

$$x=\frac{-\left(-\frac{5}{8}\right)±\frac{1}{2}}{2\times \frac{1}{64}}$$

The opposite of -\frac{5}{8} is \frac{5}{8}.

$$x=\frac{\frac{5}{8}±\frac{1}{2}}{2\times \frac{1}{64}}$$

Multiply 2 times \frac{1}{64}.

$$x=\frac{\frac{5}{8}±\frac{1}{2}}{\frac{1}{32}}$$

Now solve the equation x=\frac{\frac{5}{8}±\frac{1}{2}}{\frac{1}{32}} when ± is plus. Add \frac{5}{8} to \frac{1}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{9}{8}}{\frac{1}{32}}$$

Divide \frac{9}{8} by \frac{1}{32} by multiplying \frac{9}{8} by the reciprocal of \frac{1}{32}.

$$x=36$$

Now solve the equation x=\frac{\frac{5}{8}±\frac{1}{2}}{\frac{1}{32}} when ± is minus. Subtract \frac{1}{2} from \frac{5}{8} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{1}{8}}{\frac{1}{32}}$$

Divide \frac{1}{8} by \frac{1}{32} by multiplying \frac{1}{8} by the reciprocal of \frac{1}{32}.

$$x=4$$

The equation is now solved.

$$x=36 x=4$$

Substitute 36 for x in the equation \frac{1}{8}x+\frac{3}{2}=\sqrt{x}.

$$\frac{1}{8}\times 36+\frac{3}{2}=\sqrt{36}$$

Simplify. The value x=36 satisfies the equation.

$$6=6$$

Substitute 4 for x in the equation \frac{1}{8}x+\frac{3}{2}=\sqrt{x}.

$$\frac{1}{8}\times 4+\frac{3}{2}=\sqrt{4}$$

Simplify. The value x=4 satisfies the equation.

$$2=2$$

List all solutions of \frac{x}{8}+\frac{3}{2}=\sqrt{x}.

$$x=36 x=4$$