Solve 4k^2-12k+13=(2k-3)^2+4

$$4 k ^ { 2 } - 12 k + 13 = ( 2 k - 3 ) ^ { 2 } + 4$$

Solve for k (complex solution)

$$k\in \mathrm{C}$$

Solution Steps

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

$$4k^{2}-12k+13=4k^{2}-12k+9+4$$

Add 9 and 4 to get 13.

$$4k^{2}-12k+13=4k^{2}-12k+13$$

Subtract 4k^{2} from both sides.

$$4k^{2}-12k+13-4k^{2}=-12k+13$$

Combine 4k^{2} and -4k^{2} to get 0.

$$-12k+13=-12k+13$$

Add 12k to both sides.

$$-12k+13+12k=13$$

Combine -12k and 12k to get 0.

$$13=13$$

Compare 13 and 13.

$$\text{true}$$

This is true for any k.

$$k\in \mathrm{C}$$

Solve for k

$$k\in \mathrm{R}$$

Solution Steps

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

$$4k^{2}-12k+13=4k^{2}-12k+9+4$$

Add 9 and 4 to get 13.

$$4k^{2}-12k+13=4k^{2}-12k+13$$

Subtract 4k^{2} from both sides.

$$4k^{2}-12k+13-4k^{2}=-12k+13$$

Combine 4k^{2} and -4k^{2} to get 0.

$$-12k+13=-12k+13$$

Add 12k to both sides.

$$-12k+13+12k=13$$

Combine -12k and 12k to get 0.

$$13=13$$

Compare 13 and 13.

$$\text{true}$$

This is true for any k.

$$k\in \mathrm{R}$$