Solve sqrt{x+3}=5x+1

$$\sqrt { x + 3 } = 5 x + 1$$

Solve for x

$$x=\frac{\sqrt{281}-9}{50}\approx 0.155261092$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x+3}\right)^{2}=\left(5x+1\right)^{2}$$

Calculate \sqrt{x+3} to the power of 2 and get x+3.

$$x+3=\left(5x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+1\right)^{2}.

$$x+3=25x^{2}+10x+1$$

Subtract 25x^{2} from both sides.

$$x+3-25x^{2}=10x+1$$

Subtract 10x from both sides.

$$x+3-25x^{2}-10x=1$$

Combine x and -10x to get -9x.

$$-9x+3-25x^{2}=1$$

Subtract 1 from both sides.

$$-9x+3-25x^{2}-1=0$$

Subtract 1 from 3 to get 2.

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

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.

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

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

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

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-25\right)\times 2}}{2\left(-25\right)}$$

Multiply -4 times -25.

$$x=\frac{-\left(-9\right)±\sqrt{81+100\times 2}}{2\left(-25\right)}$$

Multiply 100 times 2.

$$x=\frac{-\left(-9\right)±\sqrt{81+200}}{2\left(-25\right)}$$

Add 81 to 200.

$$x=\frac{-\left(-9\right)±\sqrt{281}}{2\left(-25\right)}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{281}}{2\left(-25\right)}$$

Multiply 2 times -25.

$$x=\frac{9±\sqrt{281}}{-50}$$

Now solve the equation x=\frac{9±\sqrt{281}}{-50} when ± is plus. Add 9 to \sqrt{281}.

$$x=\frac{\sqrt{281}+9}{-50}$$

Divide 9+\sqrt{281} by -50.

$$x=\frac{-\sqrt{281}-9}{50}$$

Now solve the equation x=\frac{9±\sqrt{281}}{-50} when ± is minus. Subtract \sqrt{281} from 9.

$$x=\frac{9-\sqrt{281}}{-50}$$

Divide 9-\sqrt{281} by -50.

$$x=\frac{\sqrt{281}-9}{50}$$

The equation is now solved.

$$x=\frac{-\sqrt{281}-9}{50} x=\frac{\sqrt{281}-9}{50}$$

Substitute \frac{-\sqrt{281}-9}{50} for x in the equation \sqrt{x+3}=5x+1.

$$\sqrt{\frac{-\sqrt{281}-9}{50}+3}=5\times \frac{-\sqrt{281}-9}{50}+1$$

Simplify. The value x=\frac{-\sqrt{281}-9}{50} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\left(\frac{1}{10}-\frac{1}{10}\times 281^{\frac{1}{2}}\right)=-\frac{1}{10}\times 281^{\frac{1}{2}}+\frac{1}{10}$$

Substitute \frac{\sqrt{281}-9}{50} for x in the equation \sqrt{x+3}=5x+1.

$$\sqrt{\frac{\sqrt{281}-9}{50}+3}=5\times \frac{\sqrt{281}-9}{50}+1$$

Simplify. The value x=\frac{\sqrt{281}-9}{50} satisfies the equation.

$$\frac{1}{10}+\frac{1}{10}\times 281^{\frac{1}{2}}=\frac{1}{10}\times 281^{\frac{1}{2}}+\frac{1}{10}$$

Equation \sqrt{x+3}=5x+1 has a unique solution.

$$x=\frac{\sqrt{281}-9}{50}$$