Solve a-sqrt{a}=4

$$a - \sqrt { a } = 4$$

Solve for a

$$a = \frac{\sqrt{17} + 9}{2} \approx 6.561552813$$

Solution Steps

Subtract a from both sides of the equation.

$$-\sqrt{a}=4-a$$

Square both sides of the equation.

$$\left(-\sqrt{a}\right)^{2}=\left(4-a\right)^{2}$$

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

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

Calculate -1 to the power of 2 and get 1.

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

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

$$1a=\left(4-a\right)^{2}$$

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

$$1a=16-8a+a^{2}$$

Reorder the terms.

$$a=a^{2}-8a+16$$

Subtract a^{2} from both sides.

$$a-a^{2}=-8a+16$$

Add 8a to both sides.

$$a-a^{2}+8a=16$$

Combine a and 8a to get 9a.

$$9a-a^{2}=16$$

Subtract 16 from both sides.

$$9a-a^{2}-16=0$$

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.

$$-a^{2}+9a-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 9 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

Square 9.

$$a=\frac{-9±\sqrt{81-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-9±\sqrt{81+4\left(-16\right)}}{2\left(-1\right)}$$

Multiply 4 times -16.

$$a=\frac{-9±\sqrt{81-64}}{2\left(-1\right)}$$

Add 81 to -64.

$$a=\frac{-9±\sqrt{17}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{-9±\sqrt{17}}{-2}$$

Now solve the equation a=\frac{-9±\sqrt{17}}{-2} when ± is plus. Add -9 to \sqrt{17}.

$$a=\frac{\sqrt{17}-9}{-2}$$

Divide -9+\sqrt{17} by -2.

$$a=\frac{9-\sqrt{17}}{2}$$

Now solve the equation a=\frac{-9±\sqrt{17}}{-2} when ± is minus. Subtract \sqrt{17} from -9.

$$a=\frac{-\sqrt{17}-9}{-2}$$

Divide -9-\sqrt{17} by -2.

$$a=\frac{\sqrt{17}+9}{2}$$

The equation is now solved.

$$a=\frac{9-\sqrt{17}}{2} a=\frac{\sqrt{17}+9}{2}$$

Substitute \frac{9-\sqrt{17}}{2} for a in the equation a-\sqrt{a}=4.

$$\frac{9-\sqrt{17}}{2}-\sqrt{\frac{9-\sqrt{17}}{2}}=4$$

Simplify. The value a=\frac{9-\sqrt{17}}{2} does not satisfy the equation.

$$5-17^{\frac{1}{2}}=4$$

Substitute \frac{\sqrt{17}+9}{2} for a in the equation a-\sqrt{a}=4.

$$\frac{\sqrt{17}+9}{2}-\sqrt{\frac{\sqrt{17}+9}{2}}=4$$

Simplify. The value a=\frac{\sqrt{17}+9}{2} satisfies the equation.

$$4=4$$

Equation -\sqrt{a}=4-a has a unique solution.

$$a=\frac{\sqrt{17}+9}{2}$$