Solve sqrt{5x+12}=x+3

$$\sqrt{ 5x+12 } =x+3$$

Solve for x

$$x = \frac{\sqrt{13} - 1}{2} \approx 1.302775638$$

$$x=\frac{-\sqrt{13}-1}{2}\approx -2.302775638$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{5x+12}\right)^{2}=\left(x+3\right)^{2}$$

Calculate \sqrt{5x+12} to the power of 2 and get 5x+12.

$$5x+12=\left(x+3\right)^{2}$$

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

$$5x+12=x^{2}+6x+9$$

Subtract x^{2} from both sides.

$$5x+12-x^{2}=6x+9$$

Subtract 6x from both sides.

$$5x+12-x^{2}-6x=9$$

Combine 5x and -6x to get -x.

$$-x+12-x^{2}=9$$

Subtract 9 from both sides.

$$-x+12-x^{2}-9=0$$

Subtract 9 from 12 to get 3.

$$-x+3-x^{2}=0$$

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.

$$-x^{2}-x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -1 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-1\right)±\sqrt{1+4\times 3}}{2\left(-1\right)}$$

Multiply 4 times 3.

$$x=\frac{-\left(-1\right)±\sqrt{1+12}}{2\left(-1\right)}$$

Add 1 to 12.

$$x=\frac{-\left(-1\right)±\sqrt{13}}{2\left(-1\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{13}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{1±\sqrt{13}}{-2}$$

Now solve the equation x=\frac{1±\sqrt{13}}{-2} when ± is plus. Add 1 to \sqrt{13}.

$$x=\frac{\sqrt{13}+1}{-2}$$

Divide 1+\sqrt{13} by -2.

$$x=\frac{-\sqrt{13}-1}{2}$$

Now solve the equation x=\frac{1±\sqrt{13}}{-2} when ± is minus. Subtract \sqrt{13} from 1.

$$x=\frac{1-\sqrt{13}}{-2}$$

Divide 1-\sqrt{13} by -2.

$$x=\frac{\sqrt{13}-1}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{13}-1}{2} x=\frac{\sqrt{13}-1}{2}$$

Substitute \frac{-\sqrt{13}-1}{2} for x in the equation \sqrt{5x+12}=x+3.

$$\sqrt{5\times \frac{-\sqrt{13}-1}{2}+12}=\frac{-\sqrt{13}-1}{2}+3$$

Simplify. The value x=\frac{-\sqrt{13}-1}{2} satisfies the equation.

$$\frac{5}{2}-\frac{1}{2}\times 13^{\frac{1}{2}}=-\frac{1}{2}\times 13^{\frac{1}{2}}+\frac{5}{2}$$

Substitute \frac{\sqrt{13}-1}{2} for x in the equation \sqrt{5x+12}=x+3.

$$\sqrt{5\times \frac{\sqrt{13}-1}{2}+12}=\frac{\sqrt{13}-1}{2}+3$$

Simplify. The value x=\frac{\sqrt{13}-1}{2} satisfies the equation.

$$\frac{5}{2}+\frac{1}{2}\times 13^{\frac{1}{2}}=\frac{1}{2}\times 13^{\frac{1}{2}}+\frac{5}{2}$$

List all solutions of \sqrt{5x+12}=x+3.

$$x=\frac{-\sqrt{13}-1}{2} x=\frac{\sqrt{13}-1}{2}$$