Solve sqrt{4x+1}=52x

$$\sqrt { 4 x + 1 } = 52 x$$

Solve for x

$$x=\frac{\sqrt{677}+1}{1352}\approx 0.019984633$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{4x+1}\right)^{2}=\left(52x\right)^{2}$$

Calculate \sqrt{4x+1} to the power of 2 and get 4x+1.

$$4x+1=\left(52x\right)^{2}$$

Expand \left(52x\right)^{2}.

$$4x+1=52^{2}x^{2}$$

Calculate 52 to the power of 2 and get 2704.

$$4x+1=2704x^{2}$$

Subtract 2704x^{2} from both sides.

$$4x+1-2704x^{2}=0$$

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.

$$-2704x^{2}+4x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2704 for a, 4 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-2704\right)}}{2\left(-2704\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-2704\right)}}{2\left(-2704\right)}$$

Multiply -4 times -2704.

$$x=\frac{-4±\sqrt{16+10816}}{2\left(-2704\right)}$$

Add 16 to 10816.

$$x=\frac{-4±\sqrt{10832}}{2\left(-2704\right)}$$

Take the square root of 10832.

$$x=\frac{-4±4\sqrt{677}}{2\left(-2704\right)}$$

Multiply 2 times -2704.

$$x=\frac{-4±4\sqrt{677}}{-5408}$$

Now solve the equation x=\frac{-4±4\sqrt{677}}{-5408} when ± is plus. Add -4 to 4\sqrt{677}.

$$x=\frac{4\sqrt{677}-4}{-5408}$$

Divide -4+4\sqrt{677} by -5408.

$$x=\frac{1-\sqrt{677}}{1352}$$

Now solve the equation x=\frac{-4±4\sqrt{677}}{-5408} when ± is minus. Subtract 4\sqrt{677} from -4.

$$x=\frac{-4\sqrt{677}-4}{-5408}$$

Divide -4-4\sqrt{677} by -5408.

$$x=\frac{\sqrt{677}+1}{1352}$$

The equation is now solved.

$$x=\frac{1-\sqrt{677}}{1352} x=\frac{\sqrt{677}+1}{1352}$$

Substitute \frac{1-\sqrt{677}}{1352} for x in the equation \sqrt{4x+1}=52x.

$$\sqrt{4\times \frac{1-\sqrt{677}}{1352}+1}=52\times \frac{1-\sqrt{677}}{1352}$$

Simplify. The value x=\frac{1-\sqrt{677}}{1352} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\left(\frac{1}{26}-\frac{1}{26}\times 677^{\frac{1}{2}}\right)=\frac{1}{26}-\frac{1}{26}\times 677^{\frac{1}{2}}$$

Substitute \frac{\sqrt{677}+1}{1352} for x in the equation \sqrt{4x+1}=52x.

$$\sqrt{4\times \frac{\sqrt{677}+1}{1352}+1}=52\times \frac{\sqrt{677}+1}{1352}$$

Simplify. The value x=\frac{\sqrt{677}+1}{1352} satisfies the equation.

$$\frac{1}{26}+\frac{1}{26}\times 677^{\frac{1}{2}}=\frac{1}{26}\times 677^{\frac{1}{2}}+\frac{1}{26}$$

Equation \sqrt{4x+1}=52x has a unique solution.

$$x=\frac{\sqrt{677}+1}{1352}$$