Solve a+sqrt{a+1}=2

$$a + \sqrt { a + 1 } = 2$$

Solve for a

$$a=\frac{5-\sqrt{13}}{2}\approx 0.697224362$$

Solution Steps

Subtract a from both sides of the equation.

$$\sqrt{a+1}=2-a$$

Square both sides of the equation.

$$\left(\sqrt{a+1}\right)^{2}=\left(2-a\right)^{2}$$

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

$$a+1=\left(2-a\right)^{2}$$

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

$$a+1=4-4a+a^{2}$$

Subtract 4 from both sides.

$$a+1-4=-4a+a^{2}$$

Subtract 4 from 1 to get -3.

$$a-3=-4a+a^{2}$$

Add 4a to both sides.

$$a-3+4a=a^{2}$$

Combine a and 4a to get 5a.

$$5a-3=a^{2}$$

Subtract a^{2} from both sides.

$$5a-3-a^{2}=0$$

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.

$$-a^{2}+5a-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 5 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Square 5.

$$a=\frac{-5±\sqrt{25-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-5±\sqrt{25+4\left(-3\right)}}{2\left(-1\right)}$$

Multiply 4 times -3.

$$a=\frac{-5±\sqrt{25-12}}{2\left(-1\right)}$$

Add 25 to -12.

$$a=\frac{-5±\sqrt{13}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{-5±\sqrt{13}}{-2}$$

Now solve the equation a=\frac{-5±\sqrt{13}}{-2} when ± is plus. Add -5 to \sqrt{13}.

$$a=\frac{\sqrt{13}-5}{-2}$$

Divide -5+\sqrt{13} by -2.

$$a=\frac{5-\sqrt{13}}{2}$$

Now solve the equation a=\frac{-5±\sqrt{13}}{-2} when ± is minus. Subtract \sqrt{13} from -5.

$$a=\frac{-\sqrt{13}-5}{-2}$$

Divide -5-\sqrt{13} by -2.

$$a=\frac{\sqrt{13}+5}{2}$$

The equation is now solved.

$$a=\frac{5-\sqrt{13}}{2} a=\frac{\sqrt{13}+5}{2}$$

Substitute \frac{5-\sqrt{13}}{2} for a in the equation a+\sqrt{a+1}=2.

$$\frac{5-\sqrt{13}}{2}+\sqrt{\frac{5-\sqrt{13}}{2}+1}=2$$

Simplify. The value a=\frac{5-\sqrt{13}}{2} satisfies the equation.

$$2=2$$

Substitute \frac{\sqrt{13}+5}{2} for a in the equation a+\sqrt{a+1}=2.

$$\frac{\sqrt{13}+5}{2}+\sqrt{\frac{\sqrt{13}+5}{2}+1}=2$$

Simplify. The value a=\frac{\sqrt{13}+5}{2} does not satisfy the equation.

$$13^{\frac{1}{2}}+3=2$$

Equation \sqrt{a+1}=2-a has a unique solution.

$$a=\frac{5-\sqrt{13}}{2}$$