Solve 36(x-1/4)^2-64(x-5/4)^2

$$36 ( x - \frac { 1 } { 4 } ) ^ { 2 } - 64 ( x - \frac { 5 } { 4 } ) ^ { 2 }$$

Evaluate

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

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{1}{4}\right)^{2}.

$$36\left(x^{2}-\frac{1}{2}x+\frac{1}{16}\right)-64\left(x-\frac{5}{4}\right)^{2}$$

Use the distributive property to multiply 36 by x^{2}-\frac{1}{2}x+\frac{1}{16}.

$$36x^{2}-18x+\frac{9}{4}-64\left(x-\frac{5}{4}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{5}{4}\right)^{2}.

$$36x^{2}-18x+\frac{9}{4}-64\left(x^{2}-\frac{5}{2}x+\frac{25}{16}\right)$$

Use the distributive property to multiply -64 by x^{2}-\frac{5}{2}x+\frac{25}{16}.

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

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 binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{1}{4}\right)^{2}.

$$36\left(x^{2}-\frac{1}{2}x+\frac{1}{16}\right)-64\left(x-\frac{5}{4}\right)^{2}$$

Use the distributive property to multiply 36 by x^{2}-\frac{1}{2}x+\frac{1}{16}.

$$36x^{2}-18x+\frac{9}{4}-64\left(x-\frac{5}{4}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{5}{4}\right)^{2}.

$$36x^{2}-18x+\frac{9}{4}-64\left(x^{2}-\frac{5}{2}x+\frac{25}{16}\right)$$

Use the distributive property to multiply -64 by x^{2}-\frac{5}{2}x+\frac{25}{16}.

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

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}.

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