Solve sqrt{x}=x+3/4

$$\sqrt { x } = \frac { x + 3 } { 4 }$$

Solve for x

$$x=9$$

$$x=1$$

Solution Steps

Square both sides of the equation.

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

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

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

To raise \frac{x+3}{4} to a power, raise both numerator and denominator to the power and then divide.

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

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

$$x=\frac{x^{2}+6x+9}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$x=\frac{x^{2}+6x+9}{16}$$

Divide each term of x^{2}+6x+9 by 16 to get \frac{1}{16}x^{2}+\frac{3}{8}x+\frac{9}{16}.

$$x=\frac{1}{16}x^{2}+\frac{3}{8}x+\frac{9}{16}$$

Subtract \frac{1}{16}x^{2} from both sides.

$$x-\frac{1}{16}x^{2}=\frac{3}{8}x+\frac{9}{16}$$

Subtract \frac{3}{8}x from both sides.

$$x-\frac{1}{16}x^{2}-\frac{3}{8}x=\frac{9}{16}$$

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

$$\frac{5}{8}x-\frac{1}{16}x^{2}=\frac{9}{16}$$

Subtract \frac{9}{16} from both sides.

$$\frac{5}{8}x-\frac{1}{16}x^{2}-\frac{9}{16}=0$$

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.

$$-\frac{1}{16}x^{2}+\frac{5}{8}x-\frac{9}{16}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{16} for a, \frac{5}{8} for b, and -\frac{9}{16} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

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

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

Multiply -4 times -\frac{1}{16}.

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

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

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

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.

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

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

$$x=\frac{-\frac{5}{8}±\frac{1}{2}}{2\left(-\frac{1}{16}\right)}$$

Multiply 2 times -\frac{1}{16}.

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

Now solve the equation x=\frac{-\frac{5}{8}±\frac{1}{2}}{-\frac{1}{8}} 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.

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

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

$$x=1$$

Now solve the equation x=\frac{-\frac{5}{8}±\frac{1}{2}}{-\frac{1}{8}} 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.

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

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

$$x=9$$

The equation is now solved.

$$x=1 x=9$$

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

$$\sqrt{1}=\frac{1+3}{4}$$

Simplify. The value x=1 satisfies the equation.

$$1=1$$

Substitute 9 for x in the equation \sqrt{x}=\frac{x+3}{4}.

$$\sqrt{9}=\frac{9+3}{4}$$

Simplify. The value x=9 satisfies the equation.

$$3=3$$

List all solutions of \sqrt{x}=\frac{x+3}{4}.

$$x=1 x=9$$