Solve 1/2left(x-1right)left(3x+1right)-3left(x-5right)left(x+2right)3

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

Evaluate

$$-\frac{15x^{2}}{2}+26x+\frac{179}{2}$$

Solution Steps

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

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

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

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

Apply the distributive property by multiplying each term of \frac{1}{2}x-\frac{1}{2} by each term of 3x+1.

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

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

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

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

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

Express -\frac{1}{2}\times 3 as a single fraction.

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

Fraction \frac{-3}{2} can be rewritten as -\frac{3}{2} by extracting the negative sign.

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

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

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

Multiply 3 and 3 to get 9.

$$\frac{3}{2}x^{2}-x-\frac{1}{2}-9\left(x-5\right)\left(x+2\right)$$

Use the distributive property to multiply -9 by x-5.

$$\frac{3}{2}x^{2}-x-\frac{1}{2}+\left(-9x+45\right)\left(x+2\right)$$

Apply the distributive property by multiplying each term of -9x+45 by each term of x+2.

$$\frac{3}{2}x^{2}-x-\frac{1}{2}-9x^{2}-18x+45x+90$$

Combine -18x and 45x to get 27x.

$$\frac{3}{2}x^{2}-x-\frac{1}{2}-9x^{2}+27x+90$$

Combine \frac{3}{2}x^{2} and -9x^{2} to get -\frac{15}{2}x^{2}.

$$-\frac{15}{2}x^{2}-x-\frac{1}{2}+27x+90$$

Combine -x and 27x to get 26x.

$$-\frac{15}{2}x^{2}+26x-\frac{1}{2}+90$$

Convert 90 to fraction \frac{180}{2}.

$$-\frac{15}{2}x^{2}+26x-\frac{1}{2}+\frac{180}{2}$$

Since -\frac{1}{2} and \frac{180}{2} have the same denominator, add them by adding their numerators.

$$-\frac{15}{2}x^{2}+26x+\frac{-1+180}{2}$$

Add -1 and 180 to get 179.

$$-\frac{15}{2}x^{2}+26x+\frac{179}{2}$$

Expand

$$-\frac{15x^{2}}{2}+26x+\frac{179}{2}$$

Solution Steps

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

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

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

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

Apply the distributive property by multiplying each term of \frac{1}{2}x-\frac{1}{2} by each term of 3x+1.

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

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

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

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

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

Express -\frac{1}{2}\times 3 as a single fraction.

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

Fraction \frac{-3}{2} can be rewritten as -\frac{3}{2} by extracting the negative sign.

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

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

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

Multiply 3 and 3 to get 9.

$$\frac{3}{2}x^{2}-x-\frac{1}{2}-9\left(x-5\right)\left(x+2\right)$$

Use the distributive property to multiply -9 by x-5.

$$\frac{3}{2}x^{2}-x-\frac{1}{2}+\left(-9x+45\right)\left(x+2\right)$$

Apply the distributive property by multiplying each term of -9x+45 by each term of x+2.

$$\frac{3}{2}x^{2}-x-\frac{1}{2}-9x^{2}-18x+45x+90$$

Combine -18x and 45x to get 27x.

$$\frac{3}{2}x^{2}-x-\frac{1}{2}-9x^{2}+27x+90$$

Combine \frac{3}{2}x^{2} and -9x^{2} to get -\frac{15}{2}x^{2}.

$$-\frac{15}{2}x^{2}-x-\frac{1}{2}+27x+90$$

Combine -x and 27x to get 26x.

$$-\frac{15}{2}x^{2}+26x-\frac{1}{2}+90$$

Convert 90 to fraction \frac{180}{2}.

$$-\frac{15}{2}x^{2}+26x-\frac{1}{2}+\frac{180}{2}$$

Since -\frac{1}{2} and \frac{180}{2} have the same denominator, add them by adding their numerators.

$$-\frac{15}{2}x^{2}+26x+\frac{-1+180}{2}$$

Add -1 and 180 to get 179.

$$-\frac{15}{2}x^{2}+26x+\frac{179}{2}$$