Solve =[6(x-1/4)]^2-[8(x-5/4)]^2

$$= [ 6 ( x - \frac { 1 } { 4 } ) ] ^ { 2 } - [ 8 ( x - \frac { 5 } { 4 } ) ] ^ { 2 }$$

Evaluate

$$-28x^{2}+142x-\frac{391}{4}$$

Solution Steps

Use the distributive property to multiply 6 by x-\frac{1}{4}.

$$\left(6x-\frac{3}{2}\right)^{2}-\left(8\left(x-\frac{5}{4}\right)\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6x-\frac{3}{2}\right)^{2}.

$$36x^{2}-18x+\frac{9}{4}-\left(8\left(x-\frac{5}{4}\right)\right)^{2}$$

Use the distributive property to multiply 8 by x-\frac{5}{4}.

$$36x^{2}-18x+\frac{9}{4}-\left(8x-10\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8x-10\right)^{2}.

$$36x^{2}-18x+\frac{9}{4}-\left(64x^{2}-160x+100\right)$$

To find the opposite of 64x^{2}-160x+100, find the opposite of each term.

$$36x^{2}-18x+\frac{9}{4}-64x^{2}+160x-100$$

Combine 36x^{2} and -64x^{2} to get -28x^{2}.

$$-28x^{2}-18x+\frac{9}{4}+160x-100$$

Combine -18x and 160x to get 142x.

$$-28x^{2}+142x+\frac{9}{4}-100$$

Subtract 100 from \frac{9}{4} to get -\frac{391}{4}.

$$-28x^{2}+142x-\frac{391}{4}$$

Expand

$$-28x^{2}+142x-\frac{391}{4}$$

Solution Steps

Use the distributive property to multiply 6 by x-\frac{1}{4}.

$$\left(6x-\frac{3}{2}\right)^{2}-\left(8\left(x-\frac{5}{4}\right)\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6x-\frac{3}{2}\right)^{2}.

$$36x^{2}-18x+\frac{9}{4}-\left(8\left(x-\frac{5}{4}\right)\right)^{2}$$

Use the distributive property to multiply 8 by x-\frac{5}{4}.

$$36x^{2}-18x+\frac{9}{4}-\left(8x-10\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8x-10\right)^{2}.

$$36x^{2}-18x+\frac{9}{4}-\left(64x^{2}-160x+100\right)$$

To find the opposite of 64x^{2}-160x+100, find the opposite of each term.

$$36x^{2}-18x+\frac{9}{4}-64x^{2}+160x-100$$

Combine 36x^{2} and -64x^{2} to get -28x^{2}.

$$-28x^{2}-18x+\frac{9}{4}+160x-100$$

Combine -18x and 160x to get 142x.

$$-28x^{2}+142x+\frac{9}{4}-100$$

Subtract 100 from \frac{9}{4} to get -\frac{391}{4}.

$$-28x^{2}+142x-\frac{391}{4}$$