Solve 1/4x+3=4sqrt{x-2}

$$\frac{ 1 }{ 4 } x+3=4 \sqrt{ x-2 }$$

Solve for x

$$x=80\sqrt{2}+116\approx 229.13708499$$

$$x=116-80\sqrt{2}\approx 2.86291501$$

Solution Steps

Square both sides of the equation.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{1}{4}x+3\right)^{2}.

$$\frac{1}{16}x^{2}+\frac{3}{2}x+9=\left(4\sqrt{x-2}\right)^{2}$$

Expand \left(4\sqrt{x-2}\right)^{2}.

$$\frac{1}{16}x^{2}+\frac{3}{2}x+9=4^{2}\left(\sqrt{x-2}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{1}{16}x^{2}+\frac{3}{2}x+9=16\left(\sqrt{x-2}\right)^{2}$$

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

$$\frac{1}{16}x^{2}+\frac{3}{2}x+9=16\left(x-2\right)$$

Use the distributive property to multiply 16 by x-2.

$$\frac{1}{16}x^{2}+\frac{3}{2}x+9=16x-32$$

Subtract 16x from both sides.

$$\frac{1}{16}x^{2}+\frac{3}{2}x+9-16x=-32$$

Combine \frac{3}{2}x and -16x to get -\frac{29}{2}x.

$$\frac{1}{16}x^{2}-\frac{29}{2}x+9=-32$$

Add 32 to both sides.

$$\frac{1}{16}x^{2}-\frac{29}{2}x+9+32=0$$

Add 9 and 32 to get 41.

$$\frac{1}{16}x^{2}-\frac{29}{2}x+41=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{16} for a, -\frac{29}{2} for b, and 41 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{29}{2}\right)±\sqrt{\left(-\frac{29}{2}\right)^{2}-4\times \frac{1}{16}\times 41}}{2\times \frac{1}{16}}$$

Square -\frac{29}{2} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{29}{2}\right)±\sqrt{\frac{841}{4}-4\times \frac{1}{16}\times 41}}{2\times \frac{1}{16}}$$

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

$$x=\frac{-\left(-\frac{29}{2}\right)±\sqrt{\frac{841}{4}-\frac{1}{4}\times 41}}{2\times \frac{1}{16}}$$

Multiply -\frac{1}{4} times 41.

$$x=\frac{-\left(-\frac{29}{2}\right)±\sqrt{\frac{841-41}{4}}}{2\times \frac{1}{16}}$$

Add \frac{841}{4} to -\frac{41}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{29}{2}\right)±\sqrt{200}}{2\times \frac{1}{16}}$$

Take the square root of 200.

$$x=\frac{-\left(-\frac{29}{2}\right)±10\sqrt{2}}{2\times \frac{1}{16}}$$

The opposite of -\frac{29}{2} is \frac{29}{2}.

$$x=\frac{\frac{29}{2}±10\sqrt{2}}{2\times \frac{1}{16}}$$

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

$$x=\frac{\frac{29}{2}±10\sqrt{2}}{\frac{1}{8}}$$

Now solve the equation x=\frac{\frac{29}{2}±10\sqrt{2}}{\frac{1}{8}} when ± is plus. Add \frac{29}{2} to 10\sqrt{2}.

$$x=\frac{10\sqrt{2}+\frac{29}{2}}{\frac{1}{8}}$$

Divide \frac{29}{2}+10\sqrt{2} by \frac{1}{8} by multiplying \frac{29}{2}+10\sqrt{2} by the reciprocal of \frac{1}{8}.

$$x=80\sqrt{2}+116$$

Now solve the equation x=\frac{\frac{29}{2}±10\sqrt{2}}{\frac{1}{8}} when ± is minus. Subtract 10\sqrt{2} from \frac{29}{2}.

$$x=\frac{\frac{29}{2}-10\sqrt{2}}{\frac{1}{8}}$$

Divide \frac{29}{2}-10\sqrt{2} by \frac{1}{8} by multiplying \frac{29}{2}-10\sqrt{2} by the reciprocal of \frac{1}{8}.

$$x=116-80\sqrt{2}$$

The equation is now solved.

$$x=80\sqrt{2}+116 x=116-80\sqrt{2}$$

Substitute 80\sqrt{2}+116 for x in the equation \frac{1}{4}x+3=4\sqrt{x-2}.

$$\frac{1}{4}\left(80\sqrt{2}+116\right)+3=4\sqrt{80\sqrt{2}+116-2}$$

Simplify. The value x=80\sqrt{2}+116 satisfies the equation.

$$20\times 2^{\frac{1}{2}}+32=32+20\times 2^{\frac{1}{2}}$$

Substitute 116-80\sqrt{2} for x in the equation \frac{1}{4}x+3=4\sqrt{x-2}.

$$\frac{1}{4}\left(116-80\sqrt{2}\right)+3=4\sqrt{116-80\sqrt{2}-2}$$

Simplify. The value x=116-80\sqrt{2} satisfies the equation.

$$32-20\times 2^{\frac{1}{2}}=32-20\times 2^{\frac{1}{2}}$$

List all solutions of \frac{x}{4}+3=4\sqrt{x-2}.

$$x=80\sqrt{2}+116 x=116-80\sqrt{2}$$