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Question
n\left(n+1\right)+3=45
Solve the quadratic equation
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Solve by factoring
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Solve using the quadratic formula
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Solve by completing the square
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Solve using the PQ formula
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n_{1}=-7,n_{2}=6
Evaluate
n\left(n+1\right)+3=45
Expand the expression
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Evaluate
n\left(n+1\right)
Apply the distributive property
n\times n+n\times 1
Multiply the terms
n^{2}+n\times 1
Any expression multiplied by 1 remains the same
n^{2}+n
n^{2}+n+3=45
Move the expression to the left side
n^{2}+n-42=0
Factor the expression
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Evaluate
n^{2}+n-42
Rewrite the expression
n^{2}+\left(7-6\right)n-42
Calculate
n^{2}+7n-6n-42
Rewrite the expression
n\times n+n\times 7-6n-6\times 7
\text{Factor out }n\text{ from the expression}
n\left(n+7\right)-6n-6\times 7
\text{Factor out }-6\text{ from the expression}
n\left(n+7\right)-6\left(n+7\right)
\text{Factor out }n+7\text{ from the expression}
\left(n-6\right)\left(n+7\right)
\left(n-6\right)\left(n+7\right)=0
When the product of factors equals 0,at least one factor is 0
\begin{align}&n-6=0\\&n+7=0\end{align}
\text{Solve the equation for }n
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Evaluate
n-6=0
Move the constant to the right-hand side and change its sign
n=0+6
Removing 0 doesn't change the value,so remove it from the expression
n=6
\begin{align}&n=6\\&n+7=0\end{align}
\text{Solve the equation for }n
More Steps
Evaluate
n+7=0
Move the constant to the right-hand side and change its sign
n=0-7
Removing 0 doesn't change the value,so remove it from the expression
n=-7
\begin{align}&n=6\\&n=-7\end{align}
Solution
n_{1}=-7,n_{2}=6
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