Solve [K(7-K)]^2-4(2K)(-3K^2)

$$[ K ( 7 - K ) ] ^ { 2 } - 4 ( 2 K ) ( - 3 K ^ { 2 } )$$

Evaluate

$$K^{2}\left(K^{2}+10K+49\right)$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\left(K\left(7-K\right)\right)^{2}-4\times 2K^{3}\left(-3\right)$$

Use the distributive property to multiply K by 7-K.

$$\left(7K-K^{2}\right)^{2}-4\times 2K^{3}\left(-3\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7K-K^{2}\right)^{2}.

$$49K^{2}-14KK^{2}+\left(K^{2}\right)^{2}-4\times 2K^{3}\left(-3\right)$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$49K^{2}-14K^{3}+\left(K^{2}\right)^{2}-4\times 2K^{3}\left(-3\right)$$

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

$$49K^{2}-14K^{3}+K^{4}-4\times 2K^{3}\left(-3\right)$$

Multiply 4 and 2 to get 8.

$$49K^{2}-14K^{3}+K^{4}-8K^{3}\left(-3\right)$$

Multiply 8 and -3 to get -24.

$$49K^{2}-14K^{3}+K^{4}-\left(-24K^{3}\right)$$

The opposite of -24K^{3} is 24K^{3}.

$$49K^{2}-14K^{3}+K^{4}+24K^{3}$$

Combine -14K^{3} and 24K^{3} to get 10K^{3}.

$$49K^{2}+10K^{3}+K^{4}$$

Expand

$$K^{4}+10K^{3}+49K^{2}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\left(K\left(7-K\right)\right)^{2}-4\times 2K^{3}\left(-3\right)$$

Use the distributive property to multiply K by 7-K.

$$\left(7K-K^{2}\right)^{2}-4\times 2K^{3}\left(-3\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7K-K^{2}\right)^{2}.

$$49K^{2}-14KK^{2}+\left(K^{2}\right)^{2}-4\times 2K^{3}\left(-3\right)$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$49K^{2}-14K^{3}+\left(K^{2}\right)^{2}-4\times 2K^{3}\left(-3\right)$$

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

$$49K^{2}-14K^{3}+K^{4}-4\times 2K^{3}\left(-3\right)$$

Multiply 4 and 2 to get 8.

$$49K^{2}-14K^{3}+K^{4}-8K^{3}\left(-3\right)$$

Multiply 8 and -3 to get -24.

$$49K^{2}-14K^{3}+K^{4}-\left(-24K^{3}\right)$$

The opposite of -24K^{3} is 24K^{3}.

$$49K^{2}-14K^{3}+K^{4}+24K^{3}$$

Combine -14K^{3} and 24K^{3} to get 10K^{3}.

$$49K^{2}+10K^{3}+K^{4}$$