Solve (2)^2=(sqrt{3b-2}-sqrt{10-b})^2

$$( 2 ) ^ { 2 } = ( \sqrt { 3 b - 2 } - \sqrt { 10 - b } ) ^ { 2 }$$

Solve for b

$$b=6$$

$$b=1$$

Solution Steps

Calculate 2 to the power of 2 and get 4.

$$4=\left(\sqrt{3b-2}-\sqrt{10-b}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{3b-2}-\sqrt{10-b}\right)^{2}.

$$4=\left(\sqrt{3b-2}\right)^{2}-2\sqrt{3b-2}\sqrt{10-b}+\left(\sqrt{10-b}\right)^{2}$$

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

$$4=3b-2-2\sqrt{3b-2}\sqrt{10-b}+\left(\sqrt{10-b}\right)^{2}$$

Calculate \sqrt{10-b} to the power of 2 and get 10-b.

$$4=3b-2-2\sqrt{3b-2}\sqrt{10-b}+10-b$$

Add -2 and 10 to get 8.

$$4=3b+8-2\sqrt{3b-2}\sqrt{10-b}-b$$

Combine 3b and -b to get 2b.

$$4=2b+8-2\sqrt{3b-2}\sqrt{10-b}$$

Swap sides so that all variable terms are on the left hand side.

$$2b+8-2\sqrt{3b-2}\sqrt{10-b}=4$$

Subtract 8 from both sides.

$$2b-2\sqrt{3b-2}\sqrt{10-b}=4-8$$

Subtract 8 from 4 to get -4.

$$2b-2\sqrt{3b-2}\sqrt{10-b}=-4$$

Subtract 2b from both sides of the equation.

$$-2\sqrt{3b-2}\sqrt{10-b}=-4-2b$$

Square both sides of the equation.

$$\left(-2\sqrt{3b-2}\sqrt{10-b}\right)^{2}=\left(-2b-4\right)^{2}$$

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

$$\left(-2\right)^{2}\left(\sqrt{3b-2}\right)^{2}\left(\sqrt{10-b}\right)^{2}=\left(-2b-4\right)^{2}$$

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

$$4\left(\sqrt{3b-2}\right)^{2}\left(\sqrt{10-b}\right)^{2}=\left(-2b-4\right)^{2}$$

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

$$4\left(3b-2\right)\left(\sqrt{10-b}\right)^{2}=\left(-2b-4\right)^{2}$$

Calculate \sqrt{10-b} to the power of 2 and get 10-b.

$$4\left(3b-2\right)\left(10-b\right)=\left(-2b-4\right)^{2}$$

Use the distributive property to multiply 4 by 3b-2.

$$\left(12b-8\right)\left(10-b\right)=\left(-2b-4\right)^{2}$$

Use the distributive property to multiply 12b-8 by 10-b and combine like terms.

$$128b-12b^{2}-80=\left(-2b-4\right)^{2}$$

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

$$128b-12b^{2}-80=4b^{2}+16b+16$$

Subtract 4b^{2} from both sides.

$$128b-12b^{2}-80-4b^{2}=16b+16$$

Combine -12b^{2} and -4b^{2} to get -16b^{2}.

$$128b-16b^{2}-80=16b+16$$

Subtract 16b from both sides.

$$128b-16b^{2}-80-16b=16$$

Combine 128b and -16b to get 112b.

$$112b-16b^{2}-80=16$$

Subtract 16 from both sides.

$$112b-16b^{2}-80-16=0$$

Subtract 16 from -80 to get -96.

$$112b-16b^{2}-96=0$$

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.

$$-16b^{2}+112b-96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 112 for b, and -96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-112±\sqrt{112^{2}-4\left(-16\right)\left(-96\right)}}{2\left(-16\right)}$$

Square 112.

$$b=\frac{-112±\sqrt{12544-4\left(-16\right)\left(-96\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$b=\frac{-112±\sqrt{12544+64\left(-96\right)}}{2\left(-16\right)}$$

Multiply 64 times -96.

$$b=\frac{-112±\sqrt{12544-6144}}{2\left(-16\right)}$$

Add 12544 to -6144.

$$b=\frac{-112±\sqrt{6400}}{2\left(-16\right)}$$

Take the square root of 6400.

$$b=\frac{-112±80}{2\left(-16\right)}$$

Multiply 2 times -16.

$$b=\frac{-112±80}{-32}$$

Now solve the equation b=\frac{-112±80}{-32} when ± is plus. Add -112 to 80.

$$b=-\frac{32}{-32}$$

Divide -32 by -32.

$$b=1$$

Now solve the equation b=\frac{-112±80}{-32} when ± is minus. Subtract 80 from -112.

$$b=-\frac{192}{-32}$$

Divide -192 by -32.

$$b=6$$

The equation is now solved.

$$b=1 b=6$$

Substitute 1 for b in the equation 2^{2}=\left(\sqrt{3b-2}-\sqrt{10-b}\right)^{2}.

$$2^{2}=\left(\sqrt{3\times 1-2}-\sqrt{10-1}\right)^{2}$$

Simplify. The value b=1 satisfies the equation.

$$4=4$$

Substitute 6 for b in the equation 2^{2}=\left(\sqrt{3b-2}-\sqrt{10-b}\right)^{2}.

$$2^{2}=\left(\sqrt{3\times 6-2}-\sqrt{10-6}\right)^{2}$$

Simplify. The value b=6 satisfies the equation.

$$4=4$$

List all solutions of -2\sqrt{10-b}\sqrt{3b-2}=-2b-4.

$$b=1 b=6$$