Solve sqrt{x}=5x+9-100

$$\sqrt { x } = 5 x + 9 - 100$$

Solve for x

$$x = \frac{\sqrt{1821} + 911}{50} \approx 19.073463532$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(5x+9-100\right)^{2}$$

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

$$x=\left(5x+9-100\right)^{2}$$

Subtract 100 from 9 to get -91.

$$x=\left(5x-91\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-91\right)^{2}.

$$x=25x^{2}-910x+8281$$

Subtract 25x^{2} from both sides.

$$x-25x^{2}=-910x+8281$$

Add 910x to both sides.

$$x-25x^{2}+910x=8281$$

Combine x and 910x to get 911x.

$$911x-25x^{2}=8281$$

Subtract 8281 from both sides.

$$911x-25x^{2}-8281=0$$

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.

$$-25x^{2}+911x-8281=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -25 for a, 911 for b, and -8281 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-911±\sqrt{911^{2}-4\left(-25\right)\left(-8281\right)}}{2\left(-25\right)}$$

Square 911.

$$x=\frac{-911±\sqrt{829921-4\left(-25\right)\left(-8281\right)}}{2\left(-25\right)}$$

Multiply -4 times -25.

$$x=\frac{-911±\sqrt{829921+100\left(-8281\right)}}{2\left(-25\right)}$$

Multiply 100 times -8281.

$$x=\frac{-911±\sqrt{829921-828100}}{2\left(-25\right)}$$

Add 829921 to -828100.

$$x=\frac{-911±\sqrt{1821}}{2\left(-25\right)}$$

Multiply 2 times -25.

$$x=\frac{-911±\sqrt{1821}}{-50}$$

Now solve the equation x=\frac{-911±\sqrt{1821}}{-50} when ± is plus. Add -911 to \sqrt{1821}.

$$x=\frac{\sqrt{1821}-911}{-50}$$

Divide -911+\sqrt{1821} by -50.

$$x=\frac{911-\sqrt{1821}}{50}$$

Now solve the equation x=\frac{-911±\sqrt{1821}}{-50} when ± is minus. Subtract \sqrt{1821} from -911.

$$x=\frac{-\sqrt{1821}-911}{-50}$$

Divide -911-\sqrt{1821} by -50.

$$x=\frac{\sqrt{1821}+911}{50}$$

The equation is now solved.

$$x=\frac{911-\sqrt{1821}}{50} x=\frac{\sqrt{1821}+911}{50}$$

Substitute \frac{911-\sqrt{1821}}{50} for x in the equation \sqrt{x}=5x+9-100.

$$\sqrt{\frac{911-\sqrt{1821}}{50}}=5\times \frac{911-\sqrt{1821}}{50}+9-100$$

Simplify. The value x=\frac{911-\sqrt{1821}}{50} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\left(\frac{1}{10}-\frac{1}{10}\times 1821^{\frac{1}{2}}\right)=\frac{1}{10}-\frac{1}{10}\times 1821^{\frac{1}{2}}$$

Substitute \frac{\sqrt{1821}+911}{50} for x in the equation \sqrt{x}=5x+9-100.

$$\sqrt{\frac{\sqrt{1821}+911}{50}}=5\times \frac{\sqrt{1821}+911}{50}+9-100$$

Simplify. The value x=\frac{\sqrt{1821}+911}{50} satisfies the equation.

$$\frac{1}{10}+\frac{1}{10}\times 1821^{\frac{1}{2}}=\frac{1}{10}\times 1821^{\frac{1}{2}}+\frac{1}{10}$$

Equation \sqrt{x}=5x-91 has a unique solution.

$$x=\frac{\sqrt{1821}+911}{50}$$