Evaluate (1/4x^2+2/3x+4)+(-1/2x^2-5/3x-1 1/2)

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

Step 1

Remove parentheses.

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

Step 2

Convert $1 \frac{1}{2}$ to an improper fraction.

A mixed number is an addition of its whole and fractional parts.

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

Add $1$ and $\frac{1}{2}$ .

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $2$ and $1$ .

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

Step 3

Simplify each term.

Combine $\frac{1}{4}$ and $x^{2}$ .

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

Combine $\frac{2}{3}$ and $x$ .

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

Combine $x^{2}$ and $\frac{1}{2}$ .

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

Combine $x$ and $\frac{5}{3}$ .

$\frac{x^{2}}{4} + \frac{2 x}{3} + 4 - \frac{x^{2}}{2} - \frac{x \cdot 5}{3} - \frac{3}{2}$

Move $5$ to the left of $x$ .

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

Step 4

Simplify terms.

Combine the numerators over the common denominator.

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

Subtract $5 x$ from $2 x$ .

$4 + \frac{x^{2}}{4} + \frac{- 3 x}{3} + \frac{- x^{2}}{2} + \frac{- 3}{2}$

Step 5

Simplify each term.

Cancel the common factor of $- 3$ and $3$ .

Factor $3$ out of $- 3 x$ .

$4 + \frac{x^{2}}{4} + \frac{3 (- x)}{3} + \frac{- x^{2}}{2} + \frac{- 3}{2}$

Cancel the common factors.

Factor $3$ out of $3$ .

$4 + \frac{x^{2}}{4} + \frac{3 (- x)}{3 (1)} + \frac{- x^{2}}{2} + \frac{- 3}{2}$

Cancel the common factor.

$4 + \frac{x^{2}}{4} + \frac{3 (- x)}{3 \cdot 1} + \frac{- x^{2}}{2} + \frac{- 3}{2}$

Rewrite the expression.

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

Divide $- x$ by $1$ .

$4 + \frac{x^{2}}{4} - x + \frac{- x^{2}}{2} + \frac{- 3}{2}$

Move the negative in front of the fraction.

$4 + \frac{x^{2}}{4} - x - \frac{x^{2}}{2} + \frac{- 3}{2}$

Move the negative in front of the fraction.

$4 + \frac{x^{2}}{4} - x - \frac{x^{2}}{2} - \frac{3}{2}$

Step 6

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x^{2}}{4} - x - \frac{x^{2}}{2} + 4 \cdot \frac{2}{2} - \frac{3}{2}$

Step 7

Combine $4$ and $\frac{2}{2}$ .

$\frac{x^{2}}{4} - x - \frac{x^{2}}{2} + \frac{4 \cdot 2}{2} - \frac{3}{2}$

Step 8

Combine the numerators over the common denominator.

$\frac{x^{2}}{4} - x - \frac{x^{2}}{2} + \frac{4 \cdot 2 - 3}{2}$

Step 9

Simplify the numerator.

Multiply $4$ by $2$ .

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

Subtract $3$ from $8$ .

$\frac{x^{2}}{4} - x - \frac{x^{2}}{2} + \frac{5}{2}$

Step 10

To write $- \frac{x^{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- x + \frac{x^{2}}{4} - \frac{x^{2}}{2} \cdot \frac{2}{2} + \frac{5}{2}$

Step 11

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x^{2}}{2}$ by $\frac{2}{2}$ .

$- x + \frac{x^{2}}{4} - \frac{x^{2} \cdot 2}{2 \cdot 2} + \frac{5}{2}$

Multiply $2$ by $2$ .

$- x + \frac{x^{2}}{4} - \frac{x^{2} \cdot 2}{4} + \frac{5}{2}$

Step 12

Combine the numerators over the common denominator.

$- x + \frac{x^{2} - x^{2} \cdot 2}{4} + \frac{5}{2}$

Step 13

Find the common denominator.

Write $- x$ as a fraction with denominator $1$ .

$\frac{- x}{1} + \frac{x^{2} - x^{2} \cdot 2}{4} + \frac{5}{2}$

Multiply $\frac{- x}{1}$ by $\frac{4}{4}$ .

$\frac{- x}{1} \cdot \frac{4}{4} + \frac{x^{2} - x^{2} \cdot 2}{4} + \frac{5}{2}$

Multiply $\frac{- x}{1}$ by $\frac{4}{4}$ .

$\frac{- x \cdot 4}{4} + \frac{x^{2} - x^{2} \cdot 2}{4} + \frac{5}{2}$

Multiply $\frac{5}{2}$ by $\frac{2}{2}$ .

$\frac{- x \cdot 4}{4} + \frac{x^{2} - x^{2} \cdot 2}{4} + \frac{5}{2} \cdot \frac{2}{2}$

Multiply $\frac{5}{2}$ by $\frac{2}{2}$ .

$\frac{- x \cdot 4}{4} + \frac{x^{2} - x^{2} \cdot 2}{4} + \frac{5 \cdot 2}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\frac{- x \cdot 4}{4} + \frac{x^{2} - x^{2} \cdot 2}{4} + \frac{5 \cdot 2}{4}$

Step 14

Combine the numerators over the common denominator.

$\frac{- x \cdot 4 + x^{2} - x^{2} \cdot 2 + 5 \cdot 2}{4}$

Step 15

Simplify each term.

Multiply $4$ by $- 1$ .

$\frac{- 4 x + x^{2} - x^{2} \cdot 2 + 5 \cdot 2}{4}$

Multiply $2$ by $- 1$ .

$\frac{- 4 x + x^{2} - 2 x^{2} + 5 \cdot 2}{4}$

Multiply $5$ by $2$ .

$\frac{- 4 x + x^{2} - 2 x^{2} + 10}{4}$

Step 16

Simplify terms.

Subtract $2 x^{2}$ from $x^{2}$ .

$\frac{- 4 x - x^{2} + 10}{4}$

Reorder terms.

$\frac{- x^{2} - 4 x + 10}{4}$

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) - 4 x + 10}{4}$

Factor $- 1$ out of $- 4 x$ .

$\frac{- (x^{2}) - (4 x) + 10}{4}$

Factor $- 1$ out of $- (x^{2}) - (4 x)$ .

$\frac{- (x^{2} + 4 x) + 10}{4}$

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{- (x^{2} + 4 x) - 1 (- 10)}{4}$

Factor $- 1$ out of $- (x^{2} + 4 x) - 1 (- 10)$ .

$\frac{- (x^{2} + 4 x - 10)}{4}$

Simplify the expression.

Rewrite $- (x^{2} + 4 x - 10)$ as $- 1 (x^{2} + 4 x - 10)$ .

$\frac{- 1 (x^{2} + 4 x - 10)}{4}$

Move the negative in front of the fraction.

$- \frac{x^{2} + 4 x - 10}{4}$

Evaluate (1/4x^2+2/3x+4)+(-1/2x^2-5/3x-1 1/2)

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