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Question

Answer

$$3p^2-5p-\dfrac{14}{2}p+1-7p^2+8\dfrac{p}{2}p+1+5p^2+7p-\dfrac{2}{2}p+1=\dfrac{10p^2-12p+6}{2}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}3p^2-5p-\frac{14}{2}p+1-7p^2+8\frac{p}{2}p+1+5p^2+7p-\frac{2}{2}p+1& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}3p^2-5p - \frac{ 14 : \color{orangered}{ 2 } }{ 2 : \color{orangered}{ 2 }} \cdot p + 1 - 7p^2 + \frac{8p}{2}p + 1 + 5p^2 + 7p - \frac{ 2 : \color{orangered}{ 2 } }{ 2 : \color{orangered}{ 2 }} \cdot p + 1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}3p^2-5p-\frac{7}{1}p+1-7p^2+\frac{8p^2}{2}+1+5p^2+7p-\frac{1}{1}p+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}3p^2-5p-7p+1-7p^2+\frac{8p^2}{2}+1+5p^2+7p-p+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}3p^2-12p+1-7p^2+\frac{8p^2}{2}+1+5p^2+7p-p+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} } }}}-4p^2-12p+1+\frac{8p^2}{2}+1+5p^2+7p-p+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle13}{\textcircled {13}} \htmlClass{explanationCircle explanationCircle14}{\textcircled {14}} } }}}\frac{-24p+2}{2}+1+5p^2+7p-p+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle15}{\textcircled {15}} \htmlClass{explanationCircle explanationCircle16}{\textcircled {16}} } }}}\frac{-24p+4}{2}+5p^2+7p-p+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle17}{\textcircled {17}} \htmlClass{explanationCircle explanationCircle18}{\textcircled {18}} } }}}\frac{10p^2-24p+4}{2}+7p-p+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle19}{\textcircled {19}} \htmlClass{explanationCircle explanationCircle20}{\textcircled {20}} } }}}\frac{10p^2-10p+4}{2}-p+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle21}{\textcircled {21}} } }}}\frac{10p^2-12p+4}{2}+1 \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle22}{\textcircled {22}} } }}}\frac{10p^2-12p+6}{2}\end{aligned} $$
①
Divide both the top and bottom numbers by $ \color{orangered}{ 2 } $.
②

Multiply $8$ by $ \dfrac{p}{2} $ to get $ \dfrac{ 8p }{ 2 } $.

Step 1: Write $ 8 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 8 \cdot \frac{p}{2} & \xlongequal{\text{Step 1}} \frac{8}{\color{red}{1}} \cdot \frac{p}{2} \xlongequal{\text{Step 2}} \frac{ 8 \cdot p }{ 1 \cdot 2 } \xlongequal{\text{Step 3}} \frac{ 8p }{ 2 } \end{aligned} $$
③
Divide both the top and bottom numbers by $ \color{orangered}{ 2 } $.
④

Multiply $ \dfrac{8p}{2} $ by $ p $ to get $ \dfrac{ 8p^2 }{ 2 } $.

Step 1: Write $ p $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{8p}{2} \cdot p & \xlongequal{\text{Step 1}} \frac{8p}{2} \cdot \frac{p}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 8p \cdot p }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 8p^2 }{ 2 } \end{aligned} $$
⑤
Remove 1 from denominator.
⑥

Multiply $ \dfrac{8p}{2} $ by $ p $ to get $ \dfrac{ 8p^2 }{ 2 } $.

Step 1: Write $ p $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{8p}{2} \cdot p & \xlongequal{\text{Step 1}} \frac{8p}{2} \cdot \frac{p}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 8p \cdot p }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 8p^2 }{ 2 } \end{aligned} $$
⑦
Remove 1 from denominator.
⑧

Combine like terms:

$$ 3p^2 \color{blue}{-5p} \color{blue}{-7p} = 3p^2 \color{blue}{-12p} $$
⑨

Multiply $ \dfrac{8p}{2} $ by $ p $ to get $ \dfrac{ 8p^2 }{ 2 } $.

Step 1: Write $ p $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{8p}{2} \cdot p & \xlongequal{\text{Step 1}} \frac{8p}{2} \cdot \frac{p}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 8p \cdot p }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 8p^2 }{ 2 } \end{aligned} $$
⑩
Remove 1 from denominator.
⑪

Combine like terms:

$$ \color{blue}{3p^2} -12p+1 \color{blue}{-7p^2} = \color{blue}{-4p^2} -12p+1 $$
⑫
Remove 1 from denominator.
⑬

Add $-4p^2-12p+1$ and $ \dfrac{8p^2}{2} $ to get $ \dfrac{ \color{purple}{ -24p+2 } }{ 2 }$.

Step 1: Write $ -4p^2-12p+1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 2 }$.

$$ \begin{aligned} -4p^2-12p+1+ \frac{8p^2}{2} & \xlongequal{\text{Step 1}} \frac{-4p^2-12p+1}{\color{red}{1}} + \frac{8p^2}{2} = \frac{ \left( -4p^2-12p+1 \right) \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} + \frac{ 8p^2 }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -8p^2-24p+2 } }{ 2 } + \frac{ \color{purple}{ 8p^2 } }{ 2 }=\frac{ \color{purple}{ -24p+2 } }{ 2 } \end{aligned} $$
⑭
Remove 1 from denominator.
⑮

Add $ \dfrac{-24p+2}{2} $ and $ 1 $ to get $ \dfrac{ \color{purple}{ -24p+4 } }{ 2 }$.

Step 1: Write $ 1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{-24p+2}{2} +1 & \xlongequal{\text{Step 1}} \frac{-24p+2}{2} + \frac{1}{\color{red}{1}} = \frac{ -24p+2 }{ 2 } + \frac{ 1 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -24p+2 } }{ 2 } + \frac{ \color{purple}{ 2 } }{ 2 }=\frac{ \color{purple}{ -24p+4 } }{ 2 } \end{aligned} $$
⑯
Remove 1 from denominator.
⑰

Add $ \dfrac{-24p+4}{2} $ and $ 5p^2 $ to get $ \dfrac{ \color{purple}{ 10p^2-24p+4 } }{ 2 }$.

Step 1: Write $ 5p^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{-24p+4}{2} +5p^2 & \xlongequal{\text{Step 1}} \frac{-24p+4}{2} + \frac{5p^2}{\color{red}{1}} = \frac{ -24p+4 }{ 2 } + \frac{ 5p^2 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -24p+4 } }{ 2 } + \frac{ \color{purple}{ 10p^2 } }{ 2 }=\frac{ \color{purple}{ 10p^2-24p+4 } }{ 2 } \end{aligned} $$
⑱
Remove 1 from denominator.
⑲

Add $ \dfrac{10p^2-24p+4}{2} $ and $ 7p $ to get $ \dfrac{ \color{purple}{ 10p^2-10p+4 } }{ 2 }$.

Step 1: Write $ 7p $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{10p^2-24p+4}{2} +7p & \xlongequal{\text{Step 1}} \frac{10p^2-24p+4}{2} + \frac{7p}{\color{red}{1}} = \frac{ 10p^2-24p+4 }{ 2 } + \frac{ 7p \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 10p^2-24p+4 } }{ 2 } + \frac{ \color{purple}{ 14p } }{ 2 }=\frac{ \color{purple}{ 10p^2-10p+4 } }{ 2 } \end{aligned} $$
⑳
Remove 1 from denominator.
⑴

Subtract $p$ from $ \dfrac{10p^2-10p+4}{2} $ to get $ \dfrac{ \color{purple}{ 10p^2-12p+4 } }{ 2 }$.

Step 1: Write $ p $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{10p^2-10p+4}{2} -p & \xlongequal{\text{Step 1}} \frac{10p^2-10p+4}{2} - \frac{p}{\color{red}{1}} = \frac{ 10p^2-10p+4 }{ 2 } - \frac{ p \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 10p^2-10p+4 } }{ 2 } - \frac{ \color{purple}{ 2p } }{ 2 }=\frac{ \color{purple}{ 10p^2-12p+4 } }{ 2 } \end{aligned} $$
⑵

Add $ \dfrac{10p^2-12p+4}{2} $ and $ 1 $ to get $ \dfrac{ \color{purple}{ 10p^2-12p+6 } }{ 2 }$.

Step 1: Write $ 1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{10p^2-12p+4}{2} +1 & \xlongequal{\text{Step 1}} \frac{10p^2-12p+4}{2} + \frac{1}{\color{red}{1}} = \frac{ 10p^2-12p+4 }{ 2 } + \frac{ 1 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 10p^2-12p+4 } }{ 2 } + \frac{ \color{purple}{ 2 } }{ 2 }=\frac{ \color{purple}{ 10p^2-12p+6 } }{ 2 } \end{aligned} $$
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