Solve sqrt{5x+2}=5x+6

$$\sqrt{ 5x+2 } =5x+6$$

Solve for x (complex solution)

$$x=\frac{-11+\sqrt{15}i}{10}\approx -1.1+0.387298335i$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{5x+2}\right)^{2}=\left(5x+6\right)^{2}$$

Calculate \sqrt{5x+2} to the power of 2 and get 5x+2.

$$5x+2=\left(5x+6\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+6\right)^{2}.

$$5x+2=25x^{2}+60x+36$$

Subtract 25x^{2} from both sides.

$$5x+2-25x^{2}=60x+36$$

Subtract 60x from both sides.

$$5x+2-25x^{2}-60x=36$$

Combine 5x and -60x to get -55x.

$$-55x+2-25x^{2}=36$$

Subtract 36 from both sides.

$$-55x+2-25x^{2}-36=0$$

Subtract 36 from 2 to get -34.

$$-55x-34-25x^{2}=0$$

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.

$$-25x^{2}-55x-34=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -25 for a, -55 for b, and -34 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-55\right)±\sqrt{\left(-55\right)^{2}-4\left(-25\right)\left(-34\right)}}{2\left(-25\right)}$$

Square -55.

$$x=\frac{-\left(-55\right)±\sqrt{3025-4\left(-25\right)\left(-34\right)}}{2\left(-25\right)}$$

Multiply -4 times -25.

$$x=\frac{-\left(-55\right)±\sqrt{3025+100\left(-34\right)}}{2\left(-25\right)}$$

Multiply 100 times -34.

$$x=\frac{-\left(-55\right)±\sqrt{3025-3400}}{2\left(-25\right)}$$

Add 3025 to -3400.

$$x=\frac{-\left(-55\right)±\sqrt{-375}}{2\left(-25\right)}$$

Take the square root of -375.

$$x=\frac{-\left(-55\right)±5\sqrt{15}i}{2\left(-25\right)}$$

The opposite of -55 is 55.

$$x=\frac{55±5\sqrt{15}i}{2\left(-25\right)}$$

Multiply 2 times -25.

$$x=\frac{55±5\sqrt{15}i}{-50}$$

Now solve the equation x=\frac{55±5\sqrt{15}i}{-50} when ± is plus. Add 55 to 5i\sqrt{15}.

$$x=\frac{55+5\sqrt{15}i}{-50}$$

Divide 55+5i\sqrt{15} by -50.

$$x=\frac{-\sqrt{15}i-11}{10}$$

Now solve the equation x=\frac{55±5\sqrt{15}i}{-50} when ± is minus. Subtract 5i\sqrt{15} from 55.

$$x=\frac{-5\sqrt{15}i+55}{-50}$$

Divide 55-5i\sqrt{15} by -50.

$$x=\frac{-11+\sqrt{15}i}{10}$$

The equation is now solved.

$$x=\frac{-\sqrt{15}i-11}{10} x=\frac{-11+\sqrt{15}i}{10}$$

Substitute \frac{-\sqrt{15}i-11}{10} for x in the equation \sqrt{5x+2}=5x+6.

$$\sqrt{5\times \frac{-\sqrt{15}i-11}{10}+2}=5\times \frac{-\sqrt{15}i-11}{10}+6$$

Simplify. The value x=\frac{-\sqrt{15}i-11}{10} does not satisfy the equation.

$$-\left(\frac{1}{2}-\frac{1}{2}i\times 15^{\frac{1}{2}}\right)=-\frac{1}{2}i\times 15^{\frac{1}{2}}+\frac{1}{2}$$

Substitute \frac{-11+\sqrt{15}i}{10} for x in the equation \sqrt{5x+2}=5x+6.

$$\sqrt{5\times \frac{-11+\sqrt{15}i}{10}+2}=5\times \frac{-11+\sqrt{15}i}{10}+6$$

Simplify. The value x=\frac{-11+\sqrt{15}i}{10} satisfies the equation.

$$\frac{1}{2}+\frac{1}{2}i\times 15^{\frac{1}{2}}=\frac{1}{2}+\frac{1}{2}i\times 15^{\frac{1}{2}}$$

Equation \sqrt{5x+2}=5x+6 has a unique solution.

$$x=\frac{-11+\sqrt{15}i}{10}$$