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

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

Solve for x

$$x=1$$

$$x=-\frac{5}{13}\approx -0.384615385$$

Solution Steps

Subtract 2x from both sides of the equation.

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

Square both sides of the equation.

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

Expand \left(3\sqrt{1-x^{2}}\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

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

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

$$9\left(1-x^{2}\right)=\left(2-2x\right)^{2}$$

Use the distributive property to multiply 9 by 1-x^{2}.

$$9-9x^{2}=\left(2-2x\right)^{2}$$

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

$$9-9x^{2}=4-8x+4x^{2}$$

Subtract 4 from both sides.

$$9-9x^{2}-4=-8x+4x^{2}$$

Subtract 4 from 9 to get 5.

$$5-9x^{2}=-8x+4x^{2}$$

Add 8x to both sides.

$$5-9x^{2}+8x=4x^{2}$$

Subtract 4x^{2} from both sides.

$$5-9x^{2}+8x-4x^{2}=0$$

Combine -9x^{2} and -4x^{2} to get -13x^{2}.

$$5-13x^{2}+8x=0$$

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

$$-13x^{2}+8x+5=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -13x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=8 ab=-13\times 5=-65$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -65.

$$-1,65 -5,13$$

Calculate the sum for each pair.

$$-1+65=64 -5+13=8$$

The solution is the pair that gives sum 8.

$$a=13 b=-5$$

Rewrite -13x^{2}+8x+5 as \left(-13x^{2}+13x\right)+\left(-5x+5\right).

$$\left(-13x^{2}+13x\right)+\left(-5x+5\right)$$

Factor out 13x in the first and 5 in the second group.

$$13x\left(-x+1\right)+5\left(-x+1\right)$$

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

$$\left(-x+1\right)\left(13x+5\right)$$

To find equation solutions, solve -x+1=0 and 13x+5=0.

$$x=1 x=-\frac{5}{13}$$

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

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

Simplify. The value x=1 satisfies the equation.

$$2=2$$

Substitute -\frac{5}{13} for x in the equation 3\sqrt{1-x^{2}}+2x=2.

$$3\sqrt{1-\left(-\frac{5}{13}\right)^{2}}+2\left(-\frac{5}{13}\right)=2$$

Simplify. The value x=-\frac{5}{13} satisfies the equation.

$$2=2$$

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

$$x=1 x=-\frac{5}{13}$$