Solve sqrt{5x-9}-3=sqrt{x+4}-2

$$\sqrt{ 5x-9 } -3 = \sqrt{ x+4 } -2$$

Solve for x

$$x=5$$

Solution Steps

Subtract -3 from both sides of the equation.

$$\sqrt{5x-9}=\sqrt{x+4}-2+3$$

Add -2 and 3 to get 1.

$$\sqrt{5x-9}=\sqrt{x+4}+1$$

Square both sides of the equation.

$$\left(\sqrt{5x-9}\right)^{2}=\left(\sqrt{x+4}+1\right)^{2}$$

Calculate \sqrt{5x-9} to the power of 2 and get 5x-9.

$$5x-9=\left(\sqrt{x+4}+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{x+4}+1\right)^{2}.

$$5x-9=\left(\sqrt{x+4}\right)^{2}+2\sqrt{x+4}+1$$

Calculate \sqrt{x+4} to the power of 2 and get x+4.

$$5x-9=x+4+2\sqrt{x+4}+1$$

Add 4 and 1 to get 5.

$$5x-9=x+5+2\sqrt{x+4}$$

Subtract x+5 from both sides of the equation.

$$5x-9-\left(x+5\right)=2\sqrt{x+4}$$

To find the opposite of x+5, find the opposite of each term.

$$5x-9-x-5=2\sqrt{x+4}$$

Combine 5x and -x to get 4x.

$$4x-9-5=2\sqrt{x+4}$$

Subtract 5 from -9 to get -14.

$$4x-14=2\sqrt{x+4}$$

Square both sides of the equation.

$$\left(4x-14\right)^{2}=\left(2\sqrt{x+4}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-14\right)^{2}.

$$16x^{2}-112x+196=\left(2\sqrt{x+4}\right)^{2}$$

Expand \left(2\sqrt{x+4}\right)^{2}.

$$16x^{2}-112x+196=2^{2}\left(\sqrt{x+4}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$16x^{2}-112x+196=4\left(\sqrt{x+4}\right)^{2}$$

Calculate \sqrt{x+4} to the power of 2 and get x+4.

$$16x^{2}-112x+196=4\left(x+4\right)$$

Use the distributive property to multiply 4 by x+4.

$$16x^{2}-112x+196=4x+16$$

Subtract 4x from both sides.

$$16x^{2}-112x+196-4x=16$$

Combine -112x and -4x to get -116x.

$$16x^{2}-116x+196=16$$

Subtract 16 from both sides.

$$16x^{2}-116x+196-16=0$$

Subtract 16 from 196 to get 180.

$$16x^{2}-116x+180=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -116 for b, and 180 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-116\right)±\sqrt{\left(-116\right)^{2}-4\times 16\times 180}}{2\times 16}$$

Square -116.

$$x=\frac{-\left(-116\right)±\sqrt{13456-4\times 16\times 180}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-116\right)±\sqrt{13456-64\times 180}}{2\times 16}$$

Multiply -64 times 180.

$$x=\frac{-\left(-116\right)±\sqrt{13456-11520}}{2\times 16}$$

Add 13456 to -11520.

$$x=\frac{-\left(-116\right)±\sqrt{1936}}{2\times 16}$$

Take the square root of 1936.

$$x=\frac{-\left(-116\right)±44}{2\times 16}$$

The opposite of -116 is 116.

$$x=\frac{116±44}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{116±44}{32}$$

Now solve the equation x=\frac{116±44}{32} when ± is plus. Add 116 to 44.

$$x=\frac{160}{32}$$

Divide 160 by 32.

$$x=5$$

Now solve the equation x=\frac{116±44}{32} when ± is minus. Subtract 44 from 116.

$$x=\frac{72}{32}$$

Reduce the fraction \frac{72}{32} to lowest terms by extracting and canceling out 8.

$$x=\frac{9}{4}$$

The equation is now solved.

$$x=5 x=\frac{9}{4}$$

Substitute 5 for x in the equation \sqrt{5x-9}-3=\sqrt{x+4}-2.

$$\sqrt{5\times 5-9}-3=\sqrt{5+4}-2$$

Simplify. The value x=5 satisfies the equation.

$$1=1$$

Substitute \frac{9}{4} for x in the equation \sqrt{5x-9}-3=\sqrt{x+4}-2.

$$\sqrt{5\times \frac{9}{4}-9}-3=\sqrt{\frac{9}{4}+4}-2$$

Simplify. The value x=\frac{9}{4} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{3}{2}=\frac{1}{2}$$

Substitute 5 for x in the equation \sqrt{5x-9}-3=\sqrt{x+4}-2.

$$\sqrt{5\times 5-9}-3=\sqrt{5+4}-2$$

Simplify. The value x=5 satisfies the equation.

$$1=1$$

Equation \sqrt{5x-9}=\sqrt{x+4}+1 has a unique solution.

$$x=5$$