Solve sqrt{x}=5x+3

$$\sqrt{ x } =5x+3$$

Solve for x (complex solution)

$$x=\frac{-29+\sqrt{59}i}{50}\approx -0.58+0.153622915i$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(5x+3\right)^{2}$$

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

$$x=\left(5x+3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+3\right)^{2}.

$$x=25x^{2}+30x+9$$

Subtract 25x^{2} from both sides.

$$x-25x^{2}=30x+9$$

Subtract 30x from both sides.

$$x-25x^{2}-30x=9$$

Combine x and -30x to get -29x.

$$-29x-25x^{2}=9$$

Subtract 9 from both sides.

$$-29x-25x^{2}-9=0$$

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.

$$-25x^{2}-29x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -25 for a, -29 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-29\right)±\sqrt{\left(-29\right)^{2}-4\left(-25\right)\left(-9\right)}}{2\left(-25\right)}$$

Square -29.

$$x=\frac{-\left(-29\right)±\sqrt{841-4\left(-25\right)\left(-9\right)}}{2\left(-25\right)}$$

Multiply -4 times -25.

$$x=\frac{-\left(-29\right)±\sqrt{841+100\left(-9\right)}}{2\left(-25\right)}$$

Multiply 100 times -9.

$$x=\frac{-\left(-29\right)±\sqrt{841-900}}{2\left(-25\right)}$$

Add 841 to -900.

$$x=\frac{-\left(-29\right)±\sqrt{-59}}{2\left(-25\right)}$$

Take the square root of -59.

$$x=\frac{-\left(-29\right)±\sqrt{59}i}{2\left(-25\right)}$$

The opposite of -29 is 29.

$$x=\frac{29±\sqrt{59}i}{2\left(-25\right)}$$

Multiply 2 times -25.

$$x=\frac{29±\sqrt{59}i}{-50}$$

Now solve the equation x=\frac{29±\sqrt{59}i}{-50} when ± is plus. Add 29 to i\sqrt{59}.

$$x=\frac{29+\sqrt{59}i}{-50}$$

Divide 29+i\sqrt{59} by -50.

$$x=\frac{-\sqrt{59}i-29}{50}$$

Now solve the equation x=\frac{29±\sqrt{59}i}{-50} when ± is minus. Subtract i\sqrt{59} from 29.

$$x=\frac{-\sqrt{59}i+29}{-50}$$

Divide 29-i\sqrt{59} by -50.

$$x=\frac{-29+\sqrt{59}i}{50}$$

The equation is now solved.

$$x=\frac{-\sqrt{59}i-29}{50} x=\frac{-29+\sqrt{59}i}{50}$$

Substitute \frac{-\sqrt{59}i-29}{50} for x in the equation \sqrt{x}=5x+3.

$$\sqrt{\frac{-\sqrt{59}i-29}{50}}=5\times \frac{-\sqrt{59}i-29}{50}+3$$

Simplify. The value x=\frac{-\sqrt{59}i-29}{50} does not satisfy the equation.

$$-\left(\frac{1}{10}-\frac{1}{10}i\times 59^{\frac{1}{2}}\right)=-\frac{1}{10}i\times 59^{\frac{1}{2}}+\frac{1}{10}$$

Substitute \frac{-29+\sqrt{59}i}{50} for x in the equation \sqrt{x}=5x+3.

$$\sqrt{\frac{-29+\sqrt{59}i}{50}}=5\times \frac{-29+\sqrt{59}i}{50}+3$$

Simplify. The value x=\frac{-29+\sqrt{59}i}{50} satisfies the equation.

$$\frac{1}{10}+\frac{1}{10}i\times 59^{\frac{1}{2}}=\frac{1}{10}+\frac{1}{10}i\times 59^{\frac{1}{2}}$$

Equation \sqrt{x}=5x+3 has a unique solution.

$$x=\frac{-29+\sqrt{59}i}{50}$$