We want to solve the given rational equation.

\frac{7 x + 3}{x ^{2} - 8 x + 1 5} + \frac{3 x}{x - 5} = \frac{1}{3 - x}

We will start by factoring the denominators to find the least common denominator (LCD). Note that the second denominator is already factored. Let's factor the first one.

x^{2} - 8 x + 15

WriteDiff

Write as a difference

x^{2} - 5 x - 3 x + 15

▼

Factor out

x & - 3

FactorOut

Factor out $x$

x (x - 5) - 3 x + 15

FactorOut

Factor out $- 3$

x (x - 5) - 3 (x - 5)

FactorOut

Factor out $(x - 5)$

(x - 5) (x - 3)

Let's factor the third denominator.

3 - x

FactorOutNegSignSwap

$a - b = - (b - a)$

- (x - 3)

We can now rewrite the given expression.

\frac{7 x + 3}{x ^{2} - 8 x + 1 5} + \frac{3 x}{x - 5} = \frac{1}{3 - x} ⇕ \frac{7 x + 3}{( x - 5 ) ( x - 3 )} + \frac{3 x}{x - 5} = \frac{1}{- ( x - 3 )}

We will solve the equation by multiplying each side by the LCD to clear denominators.

\frac{7 x + 3}{( x - 5 ) ( x - 3 )} + \frac{3 x}{x - 5} = \frac{1}{- ( x - 3 )}

MultEqn

$LHS \cdot (x - 5) (x - 3) = RHS \cdot (x - 5) (x - 3)$

(\frac{7 x + 3}{( x - 5 ) ( x - 3 )} + \frac{3 x}{x - 5}) \cdot (x - 5) (x - 3) = \frac{1}{- ( x - 3 )} \cdot (x - 5) (x - 3)

▼

Multiply

Distr

Distribute $(x - 5) (x - 3)$

\frac{7 x + 3}{( x - 5 ) ( x - 3 )} (x - 5) (x - 3) + \frac{3 x}{x - 5} (x - 5) (x - 3) = \frac{1}{- ( x - 3 )} (x - 5) (x - 3)

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{( 7 x + 3 ) ( x - 5 ) ( x - 3 )}{( x - 5 ) ( x - 3 )} + \frac{3 x ( x - 5 ) ( x - 3 )}{x - 5} = \frac{( x - 5 ) ( x - 3 )}{- ( x - 3 )}

CancelCommonFac

Cancel out common factors

\frac{( 7 x + 3 ) ( x - 5 ) ( x - 3 )}{( x - 5 ) ( x - 3 )} + \frac{3 x ( x - 5 ) ( x - 3 )}{x - 5} = \frac{( x - 5 ) ( x - 3 )}{- ( x - 3 )}

SimpQuot

Simplify quotient

7 x + 3 + 3 x (x - 3) = \frac{x - 5}{- 1}

▼

Simplify

MultEqn

$LHS \cdot (- 1) = RHS \cdot (- 1)$

- 7 x - 3 - 3 x (x - 3) = x - 5

Distr

Distribute $- 3 x$

- 7 x - 3 - 3 x^{2} + 9 x = x - 5

AddTerms

Add terms

- 3 x^{2} + 2 x - 3 = x - 5

AddEqn

$LHS + 5 = RHS + 5$

- 3 x^{2} + 2 x + 2 = x

SubEqn

$LHS - x = RHS - x$

- 3 x^{2} + x + 2 = 0

Note that we have a quadratic equation now. Let's identify the values of

a,

b,

and

c .

- 3 x^{2} + x + 2 = 0 \Leftrightarrow - 3 x^{2} + 1 x + 2 = 0

We have that

a = - 3,

b = 1,

and

c = 2 .

Let's substitute these values into the Quadratic Formula and solve for

x .

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

SubstituteValues

Substitute values

x = \frac{- 1 \pm 1 ^{2} - 4 ( - 3 ) ( 2 )}{2 ( - 3 )}

▼

Simplify right-hand side

CalcPowProd

Calculate power and product

x = \frac{- 1 \pm 1 + 2 4}{- 6}

AddTerms

Add terms

x = \frac{- 1 \pm 2 5}{- 6}

CalcRoot

Calculate root

x = \frac{- 1 \pm 5}{- 6}

Let's calculate both solutions by using the positive and negative signs.

$x = \frac{- 1 \pm 5}{- 6}$
$x = \frac{- 1 + 5}{- 6}$	$x = \frac{- 1 - 5}{- 6}$
$x = \frac{4}{- 6}$	$x = \frac{- 6}{- 6}$
$x = - \frac{2}{3}$	$x = 1$

Finally, we will check if either

x = - \frac{2}{3}

or

x = 1

is an extraneous solution. To do this, we need to substitute

- \frac{2}{3}

and

1

for

x

in the original equation. Let's start with

x = - \frac{2}{3} .

\frac{7 x + 3}{x ^{2} - 8 x + 1 5} + \frac{3 x}{x - 5} = \frac{1}{3 - x}

Substitute

$x = - \frac{2}{3}$

\frac{7 ( - \frac{2}{3} ) + 3}{( - \frac{2}{3} ) ^{2} - 8 ( - \frac{2}{3} ) + 1 5} + \frac{3 ( - \frac{2}{3} )}{- \frac{2}{3} - 5} = ? \frac{1}{3 - ( - \frac{2}{3} )}

▼

Simplify

CalcPowProd

Calculate power and product

\frac{- \frac{1 4}{3} + 3}{\frac{4}{9} + \frac{1 6}{3} + 1 5} + \frac{- 2}{- \frac{2}{3} - 5} = ? \frac{1}{3 - ( - \frac{2}{3} )}

NumberToFrac

$a = \frac{3 \cdot a}{3}$

\frac{- \frac{1 4}{3} + \frac{9}{3}}{\frac{4}{9} + \frac{1 6}{3} + 1 5} + \frac{- 2}{- \frac{2}{3} - \frac{1 5}{3}} = ? \frac{1}{\frac{9}{3} - ( - \frac{2}{3} )}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 3}{b \cdot 3}$

\frac{- \frac{1 4}{3} + \frac{9}{3}}{\frac{4}{9} + \frac{4 8}{9} + 1 5} + \frac{- 2}{- \frac{2}{3} - \frac{1 5}{3}} = ? \frac{1}{\frac{9}{3} - ( - \frac{2}{3} )}

NumberToFrac

$a = \frac{9 \cdot a}{9}$

\frac{- \frac{1 4}{3} + \frac{9}{3}}{\frac{4}{9} + \frac{4 8}{9} + \frac{1 3 5}{9}} + \frac{- 2}{- \frac{2}{3} - \frac{1 5}{3}} = ? \frac{1}{\frac{9}{3} - ( - \frac{2}{3} )}

AddSubFrac

Add and subtract fractions

\frac{- \frac{5}{3}}{\frac{1 8 7}{9}} + \frac{- 2}{- \frac{1 7}{3}} = ? \frac{1}{\frac{1 1}{3}}

Recall that dividing by a fraction is the same as multiplying by its reciprocal.

\frac{- \frac{5}{3}}{\frac{1 8 7}{9}} + \frac{- 2}{- \frac{1 7}{3}} = ? \frac{1}{\frac{1 1}{3}} ⇕ - \frac{5}{3} (\frac{9}{1 8 7}) + (- 2) (- \frac{3}{1 7}) = ? 1 (\frac{3}{1 1})

Let's simplify to finally see whether the equation is satisfied when

x = - \frac{2}{3} .

- \frac{5}{3} (\frac{9}{1 8 7}) + (- 2) (- \frac{3}{1 7}) = ? 1 (\frac{3}{1 1})

▼

Simplify

MultFrac

Multiply fractions

\frac{- 4 5}{5 6 1} + (- 2) (- \frac{3}{1 7}) = ? 1 (\frac{3}{1 1})

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

\frac{- 4 5}{5 6 1} + \frac{6}{1 7} = ? \frac{3}{1 1}

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

\frac{- 1 5}{1 8 7} + \frac{6}{1 7} = ? \frac{3}{1 1}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 1 1}{b \cdot 1 1}$

\frac{- 1 5}{1 8 7} + \frac{6 6}{1 8 7} = ? \frac{3}{1 1}

AddFrac

Add fractions

\frac{5 1}{1 8 7} = ? \frac{3}{1 1}

ReduceFrac

$\frac{a}{b} = \frac{a / 1 7}{b / 1 7}$

\frac{3}{1 1} = \frac{3}{1 1} ✓

Now, we will check our second solution,

x = 1 .

\frac{7 x + 3}{x ^{2} - 8 x + 1 5} + \frac{3 x}{x - 5} = \frac{1}{3 - x}

Substitute

$x = 1$

\frac{7 ( 1 ) + 3}{1 ^{2} - 8 ( 1 ) + 1 5} + \frac{3 ( 1 )}{1 - 5} = ? \frac{1}{3 - 1}

▼

Simplify

CalcPowProd

Calculate power and product

\frac{7 + 3}{1 - 8 + 1 5} + \frac{3}{1 - 5} = ? \frac{1}{3 - 1}

AddSubTerms

Add and subtract terms

\frac{1 0}{8} + \frac{3}{- 4} = ? \frac{1}{2}

MoveNegDenomToFrac

Put minus sign in front of fraction

\frac{1 0}{8} - \frac{3}{4} = ? \frac{1}{2}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 2}{b \cdot 2}$

\frac{1 0}{8} - \frac{6}{8} = ? \frac{1}{2}

SubFrac

Subtract fractions

\frac{4}{8} = ? \frac{1}{2}

ReduceFrac

$\frac{a}{b} = \frac{a / 4}{b / 4}$

\frac{1}{2} = \frac{1}{2} ✓

Neither of our solutions is an extraneous solution. Therefore, the solutions to the given equation are

x = - \frac{2}{3}

and

x = 1 .

Exercise