Solve x-13sqrt{x}+40=0

$$x - 13 \sqrt { x } + 40 = 0$$

Solve for x

$$x=64$$

$$x=25$$

Solution Steps

Subtract 40 from both sides. Anything subtracted from zero gives its negation.

$$x-13\sqrt{x}=-40$$

Subtract x from both sides of the equation.

$$-13\sqrt{x}=-40-x$$

Square both sides of the equation.

$$\left(-13\sqrt{x}\right)^{2}=\left(-40-x\right)^{2}$$

Expand \left(-13\sqrt{x}\right)^{2}.

$$\left(-13\right)^{2}\left(\sqrt{x}\right)^{2}=\left(-40-x\right)^{2}$$

Calculate -13 to the power of 2 and get 169.

$$169\left(\sqrt{x}\right)^{2}=\left(-40-x\right)^{2}$$

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

$$169x=\left(-40-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-40-x\right)^{2}.

$$169x=1600+80x+x^{2}$$

Subtract 80x from both sides.

$$169x-80x=1600+x^{2}$$

Combine 169x and -80x to get 89x.

$$89x=1600+x^{2}$$

Subtract x^{2} from both sides.

$$89x-x^{2}=1600$$

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.

$$-x^{2}+89x=1600$$

Subtract 1600 from both sides of the equation.

$$-x^{2}+89x-1600=1600-1600$$

Subtracting 1600 from itself leaves 0.

$$-x^{2}+89x-1600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 89 for b, and -1600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-89±\sqrt{89^{2}-4\left(-1\right)\left(-1600\right)}}{2\left(-1\right)}$$

Square 89.

$$x=\frac{-89±\sqrt{7921-4\left(-1\right)\left(-1600\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-89±\sqrt{7921+4\left(-1600\right)}}{2\left(-1\right)}$$

Multiply 4 times -1600.

$$x=\frac{-89±\sqrt{7921-6400}}{2\left(-1\right)}$$

Add 7921 to -6400.

$$x=\frac{-89±\sqrt{1521}}{2\left(-1\right)}$$

Take the square root of 1521.

$$x=\frac{-89±39}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-89±39}{-2}$$

Now solve the equation x=\frac{-89±39}{-2} when ± is plus. Add -89 to 39.

$$x=-\frac{50}{-2}$$

Divide -50 by -2.

$$x=25$$

Now solve the equation x=\frac{-89±39}{-2} when ± is minus. Subtract 39 from -89.

$$x=-\frac{128}{-2}$$

Divide -128 by -2.

$$x=64$$

The equation is now solved.

$$x=25 x=64$$

Substitute 25 for x in the equation x-13\sqrt{x}+40=0.

$$25-13\sqrt{25}+40=0$$

Simplify. The value x=25 satisfies the equation.

$$0=0$$

Substitute 64 for x in the equation x-13\sqrt{x}+40=0.

$$64-13\sqrt{64}+40=0$$

Simplify. The value x=64 satisfies the equation.

$$0=0$$

List all solutions of -13\sqrt{x}=-x-40.

$$x=25 x=64$$