Solve 5(x+5)/20-frac{sqrt{(x-3)}}{2}=2

$$\frac { 5 ( x + 5 ) } { 20 } - \frac { \sqrt { ( x - 3 ) } } { 2 } = 2$$

Solve for x

$$x=7$$

$$x=3$$

Solution Steps

Multiply both sides of the equation by 20, the least common multiple of 20,2.

$$5\left(x+5\right)-10\sqrt{x-3}=40$$

Use the distributive property to multiply 5 by x+5.

$$5x+25-10\sqrt{x-3}=40$$

Subtract 25 from both sides.

$$5x-10\sqrt{x-3}=40-25$$

Subtract 25 from 40 to get 15.

$$5x-10\sqrt{x-3}=15$$

Subtract 5x from both sides of the equation.

$$-10\sqrt{x-3}=15-5x$$

Square both sides of the equation.

$$\left(-10\sqrt{x-3}\right)^{2}=\left(15-5x\right)^{2}$$

Expand \left(-10\sqrt{x-3}\right)^{2}.

$$\left(-10\right)^{2}\left(\sqrt{x-3}\right)^{2}=\left(15-5x\right)^{2}$$

Calculate -10 to the power of 2 and get 100.

$$100\left(\sqrt{x-3}\right)^{2}=\left(15-5x\right)^{2}$$

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

$$100\left(x-3\right)=\left(15-5x\right)^{2}$$

Use the distributive property to multiply 100 by x-3.

$$100x-300=\left(15-5x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(15-5x\right)^{2}.

$$100x-300=225-150x+25x^{2}$$

Add 150x to both sides.

$$100x-300+150x=225+25x^{2}$$

Combine 100x and 150x to get 250x.

$$250x-300=225+25x^{2}$$

Subtract 25x^{2} from both sides.

$$250x-300-25x^{2}=225$$

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.

$$-25x^{2}+250x-300=225$$

Subtract 225 from both sides of the equation.

$$-25x^{2}+250x-300-225=225-225$$

Subtracting 225 from itself leaves 0.

$$-25x^{2}+250x-300-225=0$$

Subtract 225 from -300.

$$-25x^{2}+250x-525=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -25 for a, 250 for b, and -525 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-250±\sqrt{250^{2}-4\left(-25\right)\left(-525\right)}}{2\left(-25\right)}$$

Square 250.

$$x=\frac{-250±\sqrt{62500-4\left(-25\right)\left(-525\right)}}{2\left(-25\right)}$$

Multiply -4 times -25.

$$x=\frac{-250±\sqrt{62500+100\left(-525\right)}}{2\left(-25\right)}$$

Multiply 100 times -525.

$$x=\frac{-250±\sqrt{62500-52500}}{2\left(-25\right)}$$

Add 62500 to -52500.

$$x=\frac{-250±\sqrt{10000}}{2\left(-25\right)}$$

Take the square root of 10000.

$$x=\frac{-250±100}{2\left(-25\right)}$$

Multiply 2 times -25.

$$x=\frac{-250±100}{-50}$$

Now solve the equation x=\frac{-250±100}{-50} when ± is plus. Add -250 to 100.

$$x=-\frac{150}{-50}$$

Divide -150 by -50.

$$x=3$$

Now solve the equation x=\frac{-250±100}{-50} when ± is minus. Subtract 100 from -250.

$$x=-\frac{350}{-50}$$

Divide -350 by -50.

$$x=7$$

The equation is now solved.

$$x=3 x=7$$

Substitute 3 for x in the equation \frac{5\left(x+5\right)}{20}-\frac{\sqrt{x-3}}{2}=2.

$$\frac{5\left(3+5\right)}{20}-\frac{\sqrt{3-3}}{2}=2$$

Simplify. The value x=3 satisfies the equation.

$$2=2$$

Substitute 7 for x in the equation \frac{5\left(x+5\right)}{20}-\frac{\sqrt{x-3}}{2}=2.

$$\frac{5\left(7+5\right)}{20}-\frac{\sqrt{7-3}}{2}=2$$

Simplify. The value x=7 satisfies the equation.

$$2=2$$

List all solutions of -10\sqrt{x-3}=15-5x.

$$x=3 x=7$$