Solve -1/4(x+2)(x-5)+2/7(x+2)(x-2)

$$- \frac{ 1 }{ 4 } (x+2)(x-5)+ \frac{ 2 }{ 7 } (x+2)(x-2)$$

Evaluate

$$\frac{\left(x+2\right)\left(x+19\right)}{28}$$

Solution Steps

Use the distributive property to multiply -\frac{1}{4} by x+2.

$$\left(-\frac{1}{4}x-\frac{1}{4}\times 2\right)\left(x-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Express -\frac{1}{4}\times 2 as a single fraction.

$$\left(-\frac{1}{4}x+\frac{-2}{4}\right)\left(x-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

$$\left(-\frac{1}{4}x-\frac{1}{2}\right)\left(x-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Apply the distributive property by multiplying each term of -\frac{1}{4}x-\frac{1}{2} by each term of x-5.

$$-\frac{1}{4}xx-\frac{1}{4}x\left(-5\right)-\frac{1}{2}x-\frac{1}{2}\left(-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Multiply x and x to get x^{2}.

$$-\frac{1}{4}x^{2}-\frac{1}{4}x\left(-5\right)-\frac{1}{2}x-\frac{1}{2}\left(-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Express -\frac{1}{4}\left(-5\right) as a single fraction.

$$-\frac{1}{4}x^{2}+\frac{-\left(-5\right)}{4}x-\frac{1}{2}x-\frac{1}{2}\left(-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Multiply -1 and -5 to get 5.

$$-\frac{1}{4}x^{2}+\frac{5}{4}x-\frac{1}{2}x-\frac{1}{2}\left(-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Combine \frac{5}{4}x and -\frac{1}{2}x to get \frac{3}{4}x.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x-\frac{1}{2}\left(-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Express -\frac{1}{2}\left(-5\right) as a single fraction.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{-\left(-5\right)}{2}+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Multiply -1 and -5 to get 5.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Use the distributive property to multiply \frac{2}{7} by x+2.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\left(\frac{2}{7}x+\frac{2}{7}\times 2\right)\left(x-2\right)$$

Express \frac{2}{7}\times 2 as a single fraction.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\left(\frac{2}{7}x+\frac{2\times 2}{7}\right)\left(x-2\right)$$

Multiply 2 and 2 to get 4.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\left(\frac{2}{7}x+\frac{4}{7}\right)\left(x-2\right)$$

Apply the distributive property by multiplying each term of \frac{2}{7}x+\frac{4}{7} by each term of x-2.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}xx+\frac{2}{7}x\left(-2\right)+\frac{4}{7}x+\frac{4}{7}\left(-2\right)$$

Multiply x and x to get x^{2}.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}+\frac{2}{7}x\left(-2\right)+\frac{4}{7}x+\frac{4}{7}\left(-2\right)$$

Express \frac{2}{7}\left(-2\right) as a single fraction.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}+\frac{2\left(-2\right)}{7}x+\frac{4}{7}x+\frac{4}{7}\left(-2\right)$$

Multiply 2 and -2 to get -4.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}+\frac{-4}{7}x+\frac{4}{7}x+\frac{4}{7}\left(-2\right)$$

Fraction \frac{-4}{7} can be rewritten as -\frac{4}{7} by extracting the negative sign.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}-\frac{4}{7}x+\frac{4}{7}x+\frac{4}{7}\left(-2\right)$$

Combine -\frac{4}{7}x and \frac{4}{7}x to get 0.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}+\frac{4}{7}\left(-2\right)$$

Express \frac{4}{7}\left(-2\right) as a single fraction.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}+\frac{4\left(-2\right)}{7}$$

Multiply 4 and -2 to get -8.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}+\frac{-8}{7}$$

Fraction \frac{-8}{7} can be rewritten as -\frac{8}{7} by extracting the negative sign.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}-\frac{8}{7}$$

Combine -\frac{1}{4}x^{2} and \frac{2}{7}x^{2} to get \frac{1}{28}x^{2}.

$$\frac{1}{28}x^{2}+\frac{3}{4}x+\frac{5}{2}-\frac{8}{7}$$

Least common multiple of 2 and 7 is 14. Convert \frac{5}{2} and \frac{8}{7} to fractions with denominator 14.

$$\frac{1}{28}x^{2}+\frac{3}{4}x+\frac{35}{14}-\frac{16}{14}$$

Since \frac{35}{14} and \frac{16}{14} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{28}x^{2}+\frac{3}{4}x+\frac{35-16}{14}$$

Subtract 16 from 35 to get 19.

$$\frac{1}{28}x^{2}+\frac{3}{4}x+\frac{19}{14}$$

Expand

$$\frac{x^{2}}{28}+\frac{3x}{4}+\frac{19}{14}$$

Solution Steps

Use the distributive property to multiply -\frac{1}{4} by x+2.

$$\left(-\frac{1}{4}x-\frac{1}{4}\times 2\right)\left(x-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Express -\frac{1}{4}\times 2 as a single fraction.

$$\left(-\frac{1}{4}x+\frac{-2}{4}\right)\left(x-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

$$\left(-\frac{1}{4}x-\frac{1}{2}\right)\left(x-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Apply the distributive property by multiplying each term of -\frac{1}{4}x-\frac{1}{2} by each term of x-5.

$$-\frac{1}{4}xx-\frac{1}{4}x\left(-5\right)-\frac{1}{2}x-\frac{1}{2}\left(-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Multiply x and x to get x^{2}.

$$-\frac{1}{4}x^{2}-\frac{1}{4}x\left(-5\right)-\frac{1}{2}x-\frac{1}{2}\left(-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Express -\frac{1}{4}\left(-5\right) as a single fraction.

$$-\frac{1}{4}x^{2}+\frac{-\left(-5\right)}{4}x-\frac{1}{2}x-\frac{1}{2}\left(-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Multiply -1 and -5 to get 5.

$$-\frac{1}{4}x^{2}+\frac{5}{4}x-\frac{1}{2}x-\frac{1}{2}\left(-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Combine \frac{5}{4}x and -\frac{1}{2}x to get \frac{3}{4}x.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x-\frac{1}{2}\left(-5\right)+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Express -\frac{1}{2}\left(-5\right) as a single fraction.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{-\left(-5\right)}{2}+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Multiply -1 and -5 to get 5.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}\left(x+2\right)\left(x-2\right)$$

Use the distributive property to multiply \frac{2}{7} by x+2.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\left(\frac{2}{7}x+\frac{2}{7}\times 2\right)\left(x-2\right)$$

Express \frac{2}{7}\times 2 as a single fraction.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\left(\frac{2}{7}x+\frac{2\times 2}{7}\right)\left(x-2\right)$$

Multiply 2 and 2 to get 4.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\left(\frac{2}{7}x+\frac{4}{7}\right)\left(x-2\right)$$

Apply the distributive property by multiplying each term of \frac{2}{7}x+\frac{4}{7} by each term of x-2.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}xx+\frac{2}{7}x\left(-2\right)+\frac{4}{7}x+\frac{4}{7}\left(-2\right)$$

Multiply x and x to get x^{2}.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}+\frac{2}{7}x\left(-2\right)+\frac{4}{7}x+\frac{4}{7}\left(-2\right)$$

Express \frac{2}{7}\left(-2\right) as a single fraction.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}+\frac{2\left(-2\right)}{7}x+\frac{4}{7}x+\frac{4}{7}\left(-2\right)$$

Multiply 2 and -2 to get -4.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}+\frac{-4}{7}x+\frac{4}{7}x+\frac{4}{7}\left(-2\right)$$

Fraction \frac{-4}{7} can be rewritten as -\frac{4}{7} by extracting the negative sign.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}-\frac{4}{7}x+\frac{4}{7}x+\frac{4}{7}\left(-2\right)$$

Combine -\frac{4}{7}x and \frac{4}{7}x to get 0.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}+\frac{4}{7}\left(-2\right)$$

Express \frac{4}{7}\left(-2\right) as a single fraction.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}+\frac{4\left(-2\right)}{7}$$

Multiply 4 and -2 to get -8.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}+\frac{-8}{7}$$

Fraction \frac{-8}{7} can be rewritten as -\frac{8}{7} by extracting the negative sign.

$$-\frac{1}{4}x^{2}+\frac{3}{4}x+\frac{5}{2}+\frac{2}{7}x^{2}-\frac{8}{7}$$

Combine -\frac{1}{4}x^{2} and \frac{2}{7}x^{2} to get \frac{1}{28}x^{2}.

$$\frac{1}{28}x^{2}+\frac{3}{4}x+\frac{5}{2}-\frac{8}{7}$$

Least common multiple of 2 and 7 is 14. Convert \frac{5}{2} and \frac{8}{7} to fractions with denominator 14.

$$\frac{1}{28}x^{2}+\frac{3}{4}x+\frac{35}{14}-\frac{16}{14}$$

Since \frac{35}{14} and \frac{16}{14} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{28}x^{2}+\frac{3}{4}x+\frac{35-16}{14}$$

Subtract 16 from 35 to get 19.

$$\frac{1}{28}x^{2}+\frac{3}{4}x+\frac{19}{14}$$