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Question
\frac{x^{2}+4x-45}{x^{2}+x-30}
Simplify the expression
\frac{x+9}{x+6}
Evaluate
\frac{x^{2}+4x-45}{x^{2}+x-30}
Factor the expression
More Steps
Calculate
x^{2}+4x-45
Rewrite the expression
x^{2}+\left(9-5\right)x-45
Calculate
x^{2}+9x-5x-45
Rewrite the expression
x\times x+x\times 9-5x-5\times 9
\text{Factor out }x\text{ from the expression}
x\left(x+9\right)-5x-5\times 9
\text{Factor out }-5\text{ from the expression}
x\left(x+9\right)-5\left(x+9\right)
\text{Factor out }x+9\text{ from the expression}
\left(x-5\right)\left(x+9\right)
\frac{\left(x-5\right)\left(x+9\right)}{x^{2}+x-30}
Factor the expression
More Steps
Calculate
x^{2}+x-30
Rewrite the expression
x^{2}+\left(6-5\right)x-30
Calculate
x^{2}+6x-5x-30
Rewrite the expression
x\times x+x\times 6-5x-5\times 6
\text{Factor out }x\text{ from the expression}
x\left(x+6\right)-5x-5\times 6
\text{Factor out }-5\text{ from the expression}
x\left(x+6\right)-5\left(x+6\right)
\text{Factor out }x+6\text{ from the expression}
\left(x-5\right)\left(x+6\right)
\frac{\left(x-5\right)\left(x+9\right)}{\left(x-5\right)\left(x+6\right)}
Solution
\frac{x+9}{x+6}
Show Solutions
Find the excluded values
x=5,x=-6
Evaluate
\frac{x^{2}+4x-45}{x^{2}+x-30}
To find the excluded values,set the denominators equal to 0
x^{2}+x-30=0
Factor the expression
More Steps
Evaluate
x^{2}+x-30
Rewrite the expression
x^{2}+\left(6-5\right)x-30
Calculate
x^{2}+6x-5x-30
Rewrite the expression
x\times x+x\times 6-5x-5\times 6
\text{Factor out }x\text{ from the expression}
x\left(x+6\right)-5x-5\times 6
\text{Factor out }-5\text{ from the expression}
x\left(x+6\right)-5\left(x+6\right)
\text{Factor out }x+6\text{ from the expression}
\left(x-5\right)\left(x+6\right)
\left(x-5\right)\left(x+6\right)=0
When the product of factors equals 0,at least one factor is 0
\begin{align}&x-5=0\\&x+6=0\end{align}
\text{Solve the equation for }x
More Steps
Evaluate
x-5=0
Move the constant to the right-hand side and change its sign
x=0+5
Removing 0 doesn't change the value,so remove it from the expression
x=5
\begin{align}&x=5\\&x+6=0\end{align}
\text{Solve the equation for }x
More Steps
Evaluate
x+6=0
Move the constant to the right-hand side and change its sign
x=0-6
Removing 0 doesn't change the value,so remove it from the expression
x=-6
\begin{align}&x=5\\&x=-6\end{align}
Solution
x=5,x=-6
Show Solutions
Find the roots
x=-9
Evaluate
\frac{x^{2}+4x-45}{x^{2}+x-30}
To find the roots of the expression,set the expression equal to 0
\frac{x^{2}+4x-45}{x^{2}+x-30}=0
Find the domain
More Steps
Evaluate
x^{2}+x-30\neq 0
Move the constant to the right side
x^{2}+x\neq 0-\left(-30\right)
Add the terms
x^{2}+x\neq 30
Add the same value to both sides
x^{2}+x+\frac{1}{4}\neq 30+\frac{1}{4}
Evaluate
x^{2}+x+\frac{1}{4}\neq \frac{121}{4}
Evaluate
\left(x+\frac{1}{2}\right)^{2}\neq \frac{121}{4}
Take the root of both sides of the equation and remember to use both positive and negative roots
x+\frac{1}{2}\neq \pm \sqrt{\frac{121}{4}}
Simplify the expression
More Steps
Evaluate
\sqrt{\frac{121}{4}}
To take a root of a fraction,take the root of the numerator and denominator separately
\frac{\sqrt{121}}{\sqrt{4}}
Simplify the radical expression
\frac{11}{\sqrt{4}}
Simplify the radical expression
\frac{11}{2}
x+\frac{1}{2}\neq \pm \frac{11}{2}
\text{Separate the inequality into }2\text{ possible cases}
\left\{ \begin{array}{l}x+\frac{1}{2}\neq \frac{11}{2}\\x+\frac{1}{2}\neq -\frac{11}{2}\end{array}\right.
Calculate
More Steps
Evaluate
x+\frac{1}{2}\neq \frac{11}{2}
Move the constant to the right side
x\neq \frac{11}{2}-\frac{1}{2}
Subtract the numbers
x\neq 5
\left\{ \begin{array}{l}x\neq 5\\x+\frac{1}{2}\neq -\frac{11}{2}\end{array}\right.
Calculate
More Steps
Evaluate
x+\frac{1}{2}\neq -\frac{11}{2}
Move the constant to the right side
x\neq -\frac{11}{2}-\frac{1}{2}
Subtract the numbers
x\neq -6
\left\{ \begin{array}{l}x\neq 5\\x\neq -6\end{array}\right.
Find the intersection
x \in \left(-\infty,-6\right)\cup \left(-6,5\right)\cup \left(5,+\infty\right)
\frac{x^{2}+4x-45}{x^{2}+x-30}=0,x \in \left(-\infty,-6\right)\cup \left(-6,5\right)\cup \left(5,+\infty\right)
Calculate
\frac{x^{2}+4x-45}{x^{2}+x-30}=0
Divide the terms
More Steps
Evaluate
\frac{x^{2}+4x-45}{x^{2}+x-30}
Factor the expression
More Steps
Calculate
x^{2}+4x-45
Rewrite the expression
x^{2}+\left(9-5\right)x-45
Calculate
x^{2}+9x-5x-45
Rewrite the expression
x\times x+x\times 9-5x-5\times 9
\text{Factor out }x\text{ from the expression}
x\left(x+9\right)-5x-5\times 9
\text{Factor out }-5\text{ from the expression}
x\left(x+9\right)-5\left(x+9\right)
\text{Factor out }x+9\text{ from the expression}
\left(x-5\right)\left(x+9\right)
\frac{\left(x-5\right)\left(x+9\right)}{x^{2}+x-30}
Factor the expression
More Steps
Calculate
x^{2}+x-30
Rewrite the expression
x^{2}+\left(6-5\right)x-30
Calculate
x^{2}+6x-5x-30
Rewrite the expression
x\times x+x\times 6-5x-5\times 6
\text{Factor out }x\text{ from the expression}
x\left(x+6\right)-5x-5\times 6
\text{Factor out }-5\text{ from the expression}
x\left(x+6\right)-5\left(x+6\right)
\text{Factor out }x+6\text{ from the expression}
\left(x-5\right)\left(x+6\right)
\frac{\left(x-5\right)\left(x+9\right)}{\left(x-5\right)\left(x+6\right)}
Reduce the fraction
\frac{x+9}{x+6}
\frac{x+9}{x+6}=0
Cross multiply
x+9=\left(x+6\right)\times 0
Simplify the equation
x+9=0
Move the constant to the right side
x=0-9
Removing 0 doesn't change the value,so remove it from the expression
x=-9
Check if the solution is in the defined range
x=-9,x \in \left(-\infty,-6\right)\cup \left(-6,5\right)\cup \left(5,+\infty\right)
Solution
x=-9
Show Solutions
