Solve sqrt{1-2x}=sqrt{x^2+x-3}

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

Solve for x (complex solution)

$$x=-4$$

$$x=1$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{1-2x} to the power of 2 and get 1-2x.

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

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

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

Subtract x^{2} from both sides.

$$1-2x-x^{2}=x-3$$

Subtract x from both sides.

$$1-2x-x^{2}-x=-3$$

Combine -2x and -x to get -3x.

$$1-3x-x^{2}=-3$$

Add 3 to both sides.

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

Add 1 and 3 to get 4.

$$4-3x-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}-3x+4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=-3 ab=-4=-4$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -4.

$$1,-4 2,-2$$

Calculate the sum for each pair.

$$1-4=-3 2-2=0$$

The solution is the pair that gives sum -3.

$$a=1 b=-4$$

Rewrite -x^{2}-3x+4 as \left(-x^{2}+x\right)+\left(-4x+4\right).

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

Factor out x in the first and 4 in the second group.

$$x\left(-x+1\right)+4\left(-x+1\right)$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(x+4\right)$$

To find equation solutions, solve -x+1=0 and x+4=0.

$$x=1 x=-4$$

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

$$\sqrt{1-2}=\sqrt{1^{2}+1-3}$$

Simplify. The value x=1 satisfies the equation.

$$i=i$$

Substitute -4 for x in the equation \sqrt{1-2x}=\sqrt{x^{2}+x-3}.

$$\sqrt{1-2\left(-4\right)}=\sqrt{\left(-4\right)^{2}-4-3}$$

Simplify. The value x=-4 satisfies the equation.

$$3=3$$

List all solutions of \sqrt{1-2x}=\sqrt{x^{2}+x-3}.

$$x=1 x=-4$$

Solve for x

$$x=-4$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{1-2x} to the power of 2 and get 1-2x.

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

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

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

Subtract x^{2} from both sides.

$$1-2x-x^{2}=x-3$$

Subtract x from both sides.

$$1-2x-x^{2}-x=-3$$

Combine -2x and -x to get -3x.

$$1-3x-x^{2}=-3$$

Add 3 to both sides.

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

Add 1 and 3 to get 4.

$$4-3x-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}-3x+4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=-3 ab=-4=-4$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -4.

$$1,-4 2,-2$$

Calculate the sum for each pair.

$$1-4=-3 2-2=0$$

The solution is the pair that gives sum -3.

$$a=1 b=-4$$

Rewrite -x^{2}-3x+4 as \left(-x^{2}+x\right)+\left(-4x+4\right).

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

Factor out x in the first and 4 in the second group.

$$x\left(-x+1\right)+4\left(-x+1\right)$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(x+4\right)$$

To find equation solutions, solve -x+1=0 and x+4=0.

$$x=1 x=-4$$

Substitute 1 for x in the equation \sqrt{1-2x}=\sqrt{x^{2}+x-3}. The expression \sqrt{1-2} is undefined because the radicand cannot be negative.

$$\sqrt{1-2}=\sqrt{1^{2}+1-3}$$

Substitute -4 for x in the equation \sqrt{1-2x}=\sqrt{x^{2}+x-3}.

$$\sqrt{1-2\left(-4\right)}=\sqrt{\left(-4\right)^{2}-4-3}$$

Simplify. The value x=-4 satisfies the equation.

$$3=3$$

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

$$x=-4$$