Solve (3x^2-1)^2-9x^2(1-x)^2

$$( 3 x ^ { 2 } - 1 ) ^ { 2 } - 9 x ^ { 2 } ( 1 - x ) ^ { 2 }$$

Evaluate

$$18x^{3}-15x^{2}+1$$

Solution Steps

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

$$9\left(x^{2}\right)^{2}-6x^{2}+1-9x^{2}\left(1-x\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$9x^{4}-6x^{2}+1-9x^{2}\left(1-x\right)^{2}$$

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

$$9x^{4}-6x^{2}+1-9x^{2}\left(1-2x+x^{2}\right)$$

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

$$9x^{4}-6x^{2}+1-9x^{2}+18x^{3}-9x^{4}$$

Combine -6x^{2} and -9x^{2} to get -15x^{2}.

$$9x^{4}-15x^{2}+1+18x^{3}-9x^{4}$$

Combine 9x^{4} and -9x^{4} to get 0.

$$-15x^{2}+1+18x^{3}$$

Expand

$$18x^{3}-15x^{2}+1$$

Solution Steps

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

$$9\left(x^{2}\right)^{2}-6x^{2}+1-9x^{2}\left(1-x\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$9x^{4}-6x^{2}+1-9x^{2}\left(1-x\right)^{2}$$

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

$$9x^{4}-6x^{2}+1-9x^{2}\left(1-2x+x^{2}\right)$$

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

$$9x^{4}-6x^{2}+1-9x^{2}+18x^{3}-9x^{4}$$

Combine -6x^{2} and -9x^{2} to get -15x^{2}.

$$9x^{4}-15x^{2}+1+18x^{3}-9x^{4}$$

Combine 9x^{4} and -9x^{4} to get 0.

$$-15x^{2}+1+18x^{3}$$