Solve 1/4x+2sqrt{x}+15=x

$$\frac { 1 } { 4 } x + 2 \sqrt { x } + 15 = x$$

Solve for x

$$x=36$$

Solution Steps

Subtract \frac{1}{4}x+15 from both sides of the equation.

$$2\sqrt{x}=x-\left(\frac{1}{4}x+15\right)$$

To find the opposite of \frac{1}{4}x+15, find the opposite of each term.

$$2\sqrt{x}=x-\frac{1}{4}x-15$$

Combine x and -\frac{1}{4}x to get \frac{3}{4}x.

$$2\sqrt{x}=\frac{3}{4}x-15$$

Square both sides of the equation.

$$\left(2\sqrt{x}\right)^{2}=\left(\frac{3}{4}x-15\right)^{2}$$

Expand \left(2\sqrt{x}\right)^{2}.

$$2^{2}\left(\sqrt{x}\right)^{2}=\left(\frac{3}{4}x-15\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{x}\right)^{2}=\left(\frac{3}{4}x-15\right)^{2}$$

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

$$4x=\left(\frac{3}{4}x-15\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{3}{4}x-15\right)^{2}.

$$4x=\frac{9}{16}x^{2}-\frac{45}{2}x+225$$

Subtract \frac{9}{16}x^{2} from both sides.

$$4x-\frac{9}{16}x^{2}=-\frac{45}{2}x+225$$

Add \frac{45}{2}x to both sides.

$$4x-\frac{9}{16}x^{2}+\frac{45}{2}x=225$$

Combine 4x and \frac{45}{2}x to get \frac{53}{2}x.

$$\frac{53}{2}x-\frac{9}{16}x^{2}=225$$

Subtract 225 from both sides.

$$\frac{53}{2}x-\frac{9}{16}x^{2}-225=0$$

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.

$$-\frac{9}{16}x^{2}+\frac{53}{2}x-225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{9}{16} for a, \frac{53}{2} for b, and -225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{53}{2}±\sqrt{\left(\frac{53}{2}\right)^{2}-4\left(-\frac{9}{16}\right)\left(-225\right)}}{2\left(-\frac{9}{16}\right)}$$

Square \frac{53}{2} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{53}{2}±\sqrt{\frac{2809}{4}-4\left(-\frac{9}{16}\right)\left(-225\right)}}{2\left(-\frac{9}{16}\right)}$$

Multiply -4 times -\frac{9}{16}.

$$x=\frac{-\frac{53}{2}±\sqrt{\frac{2809}{4}+\frac{9}{4}\left(-225\right)}}{2\left(-\frac{9}{16}\right)}$$

Multiply \frac{9}{4} times -225.

$$x=\frac{-\frac{53}{2}±\sqrt{\frac{2809-2025}{4}}}{2\left(-\frac{9}{16}\right)}$$

Add \frac{2809}{4} to -\frac{2025}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{53}{2}±\sqrt{196}}{2\left(-\frac{9}{16}\right)}$$

Take the square root of 196.

$$x=\frac{-\frac{53}{2}±14}{2\left(-\frac{9}{16}\right)}$$

Multiply 2 times -\frac{9}{16}.

$$x=\frac{-\frac{53}{2}±14}{-\frac{9}{8}}$$

Now solve the equation x=\frac{-\frac{53}{2}±14}{-\frac{9}{8}} when ± is plus. Add -\frac{53}{2} to 14.

$$x=-\frac{\frac{25}{2}}{-\frac{9}{8}}$$

Divide -\frac{25}{2} by -\frac{9}{8} by multiplying -\frac{25}{2} by the reciprocal of -\frac{9}{8}.

$$x=\frac{100}{9}$$

Now solve the equation x=\frac{-\frac{53}{2}±14}{-\frac{9}{8}} when ± is minus. Subtract 14 from -\frac{53}{2}.

$$x=-\frac{\frac{81}{2}}{-\frac{9}{8}}$$

Divide -\frac{81}{2} by -\frac{9}{8} by multiplying -\frac{81}{2} by the reciprocal of -\frac{9}{8}.

$$x=36$$

The equation is now solved.

$$x=\frac{100}{9} x=36$$

Substitute \frac{100}{9} for x in the equation \frac{1}{4}x+2\sqrt{x}+15=x.

$$\frac{1}{4}\times \frac{100}{9}+2\sqrt{\frac{100}{9}}+15=\frac{100}{9}$$

Simplify. The value x=\frac{100}{9} does not satisfy the equation.

$$\frac{220}{9}=\frac{100}{9}$$

Substitute 36 for x in the equation \frac{1}{4}x+2\sqrt{x}+15=x.

$$\frac{1}{4}\times 36+2\sqrt{36}+15=36$$

Simplify. The value x=36 satisfies the equation.

$$36=36$$

Equation 2\sqrt{x}=\frac{3x}{4}-15 has a unique solution.

$$x=36$$