Solve sqrt{4n+3}=n

$$\sqrt{ 4n+3 } = n$$

Solve for n

$$n=\sqrt{7}+2\approx 4.645751311$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{4n+3} to the power of 2 and get 4n+3.

$$4n+3=n^{2}$$

Subtract n^{2} from both sides.

$$4n+3-n^{2}=0$$

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.

$$-n^{2}+4n+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

Square 4.

$$n=\frac{-4±\sqrt{16-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$n=\frac{-4±\sqrt{16+4\times 3}}{2\left(-1\right)}$$

Multiply 4 times 3.

$$n=\frac{-4±\sqrt{16+12}}{2\left(-1\right)}$$

Add 16 to 12.

$$n=\frac{-4±\sqrt{28}}{2\left(-1\right)}$$

Take the square root of 28.

$$n=\frac{-4±2\sqrt{7}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{-4±2\sqrt{7}}{-2}$$

Now solve the equation n=\frac{-4±2\sqrt{7}}{-2} when ± is plus. Add -4 to 2\sqrt{7}.

$$n=\frac{2\sqrt{7}-4}{-2}$$

Divide -4+2\sqrt{7} by -2.

$$n=2-\sqrt{7}$$

Now solve the equation n=\frac{-4±2\sqrt{7}}{-2} when ± is minus. Subtract 2\sqrt{7} from -4.

$$n=\frac{-2\sqrt{7}-4}{-2}$$

Divide -4-2\sqrt{7} by -2.

$$n=\sqrt{7}+2$$

The equation is now solved.

$$n=2-\sqrt{7} n=\sqrt{7}+2$$

Substitute 2-\sqrt{7} for n in the equation \sqrt{4n+3}=n.

$$\sqrt{4\left(2-\sqrt{7}\right)+3}=2-\sqrt{7}$$

Simplify. The value n=2-\sqrt{7} does not satisfy the equation because the left and the right hand side have opposite signs.

$$7^{\frac{1}{2}}-2=2-7^{\frac{1}{2}}$$

Substitute \sqrt{7}+2 for n in the equation \sqrt{4n+3}=n.

$$\sqrt{4\left(\sqrt{7}+2\right)+3}=\sqrt{7}+2$$

Simplify. The value n=\sqrt{7}+2 satisfies the equation.

$$2+7^{\frac{1}{2}}=2+7^{\frac{1}{2}}$$

Equation \sqrt{4n+3}=n has a unique solution.

$$n=\sqrt{7}+2$$