Solve sqrt{5x}-15=3x+9

$$\sqrt{ 5x } -15 = 3x+9$$

Solve for x (complex solution)

$$x=\frac{-139+\sqrt{1415}i}{18}\approx -7.722222222+2.089804764i$$

Solution Steps

Subtract -15 from both sides of the equation.

$$\sqrt{5x}=3x+9+15$$

Add 9 and 15 to get 24.

$$\sqrt{5x}=3x+24$$

Square both sides of the equation.

$$\left(\sqrt{5x}\right)^{2}=\left(3x+24\right)^{2}$$

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

$$5x=\left(3x+24\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+24\right)^{2}.

$$5x=9x^{2}+144x+576$$

Subtract 9x^{2} from both sides.

$$5x-9x^{2}=144x+576$$

Subtract 144x from both sides.

$$5x-9x^{2}-144x=576$$

Combine 5x and -144x to get -139x.

$$-139x-9x^{2}=576$$

Subtract 576 from both sides.

$$-139x-9x^{2}-576=0$$

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.

$$-9x^{2}-139x-576=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -139 for b, and -576 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-139\right)±\sqrt{\left(-139\right)^{2}-4\left(-9\right)\left(-576\right)}}{2\left(-9\right)}$$

Square -139.

$$x=\frac{-\left(-139\right)±\sqrt{19321-4\left(-9\right)\left(-576\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-139\right)±\sqrt{19321+36\left(-576\right)}}{2\left(-9\right)}$$

Multiply 36 times -576.

$$x=\frac{-\left(-139\right)±\sqrt{19321-20736}}{2\left(-9\right)}$$

Add 19321 to -20736.

$$x=\frac{-\left(-139\right)±\sqrt{-1415}}{2\left(-9\right)}$$

Take the square root of -1415.

$$x=\frac{-\left(-139\right)±\sqrt{1415}i}{2\left(-9\right)}$$

The opposite of -139 is 139.

$$x=\frac{139±\sqrt{1415}i}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{139±\sqrt{1415}i}{-18}$$

Now solve the equation x=\frac{139±\sqrt{1415}i}{-18} when ± is plus. Add 139 to i\sqrt{1415}.

$$x=\frac{139+\sqrt{1415}i}{-18}$$

Divide 139+i\sqrt{1415} by -18.

$$x=\frac{-\sqrt{1415}i-139}{18}$$

Now solve the equation x=\frac{139±\sqrt{1415}i}{-18} when ± is minus. Subtract i\sqrt{1415} from 139.

$$x=\frac{-\sqrt{1415}i+139}{-18}$$

Divide 139-i\sqrt{1415} by -18.

$$x=\frac{-139+\sqrt{1415}i}{18}$$

The equation is now solved.

$$x=\frac{-\sqrt{1415}i-139}{18} x=\frac{-139+\sqrt{1415}i}{18}$$

Substitute \frac{-\sqrt{1415}i-139}{18} for x in the equation \sqrt{5x}-15=3x+9.

$$\sqrt{5\times \frac{-\sqrt{1415}i-139}{18}}-15=3\times \frac{-\sqrt{1415}i-139}{18}+9$$

Simplify. The value x=\frac{-\sqrt{1415}i-139}{18} does not satisfy the equation.

$$-\frac{95}{6}+\frac{1}{6}i\times 1415^{\frac{1}{2}}=-\frac{1}{6}i\times 1415^{\frac{1}{2}}-\frac{85}{6}$$

Substitute \frac{-139+\sqrt{1415}i}{18} for x in the equation \sqrt{5x}-15=3x+9.

$$\sqrt{5\times \frac{-139+\sqrt{1415}i}{18}}-15=3\times \frac{-139+\sqrt{1415}i}{18}+9$$

Simplify. The value x=\frac{-139+\sqrt{1415}i}{18} satisfies the equation.

$$-\frac{85}{6}+\frac{1}{6}i\times 1415^{\frac{1}{2}}=-\frac{85}{6}+\frac{1}{6}i\times 1415^{\frac{1}{2}}$$

Equation \sqrt{5x}=3x+24 has a unique solution.

$$x=\frac{-139+\sqrt{1415}i}{18}$$