Solve sqrt{2x-5}+sqrt{x-3}-1=0

$$\sqrt { 2 x - 5 } + \sqrt { x - 3 } - 1 = 0$$

Solve for x

$$x=3$$

Solution Steps

Subtract \sqrt{x-3}-1 from both sides of the equation.

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

To find the opposite of \sqrt{x-3}-1, find the opposite of each term.

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

The opposite of -1 is 1.

$$\sqrt{2x-5}=-\sqrt{x-3}+1$$

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-\sqrt{x-3}+1\right)^{2}.

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

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

$$2x-5=x-3-2\sqrt{x-3}+1$$

Add -3 and 1 to get -2.

$$2x-5=x-2-2\sqrt{x-3}$$

Subtract x-2 from both sides of the equation.

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

To find the opposite of x-2, find the opposite of each term.

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

Combine 2x and -x to get x.

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

Add -5 and 2 to get -3.

$$x-3=-2\sqrt{x-3}$$

Square both sides of the equation.

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

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

$$x^{2}-6x+9=\left(-2\sqrt{x-3}\right)^{2}$$

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

$$x^{2}-6x+9=\left(-2\right)^{2}\left(\sqrt{x-3}\right)^{2}$$

Calculate -2 to the power of 2 and get 4.

$$x^{2}-6x+9=4\left(\sqrt{x-3}\right)^{2}$$

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

$$x^{2}-6x+9=4\left(x-3\right)$$

Use the distributive property to multiply 4 by x-3.

$$x^{2}-6x+9=4x-12$$

Subtract 4x from both sides.

$$x^{2}-6x+9-4x=-12$$

Combine -6x and -4x to get -10x.

$$x^{2}-10x+9=-12$$

Add 12 to both sides.

$$x^{2}-10x+9+12=0$$

Add 9 and 12 to get 21.

$$x^{2}-10x+21=0$$

To solve the equation, factor x^{2}-10x+21 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-10 ab=21$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 21.

$$-1,-21 -3,-7$$

Calculate the sum for each pair.

$$-1-21=-22 -3-7=-10$$

The solution is the pair that gives sum -10.

$$a=-7 b=-3$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-7\right)\left(x-3\right)$$

To find equation solutions, solve x-7=0 and x-3=0.

$$x=7 x=3$$

Substitute 7 for x in the equation \sqrt{2x-5}+\sqrt{x-3}-1=0.

$$\sqrt{2\times 7-5}+\sqrt{7-3}-1=0$$

Simplify. The value x=7 does not satisfy the equation.

$$4=0$$

Substitute 3 for x in the equation \sqrt{2x-5}+\sqrt{x-3}-1=0.

$$\sqrt{2\times 3-5}+\sqrt{3-3}-1=0$$

Simplify. The value x=3 satisfies the equation.

$$0=0$$

Equation \sqrt{2x-5}=-\sqrt{x-3}+1 has a unique solution.

$$x=3$$