Basic Math Examples

$\frac{a^{2} + 13}{a - 8} = \frac{17 a - 59}{a - 8}$

Step 1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$a^{2} + 13 = 17 a - 59$

Step 2

Subtract $17 a$ from both sides of the equation.

$a^{2} + 13 - 17 a = - 59$

Step 3

Add $59$ to both sides of the equation.

$a^{2} + 13 - 17 a + 59 = 0$

Step 4

Add $13$ and $59$ .

$a^{2} - 17 a + 72 = 0$

Step 5

Factor $a^{2} - 17 a + 72$ using the AC method.

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Step 5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $72$ and whose sum is $- 17$ .

$- 9, - 8$

Step 5.2

Write the factored form using these integers.

$(a - 9) (a - 8) = 0$

Step 6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$a - 9 = 0$

$a - 8 = 0$

Step 7

Set $a - 9$ equal to $0$ and solve for $a$ .

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Step 7.1

Set $a - 9$ equal to $0$ .

$a - 9 = 0$

Step 7.2

Add $9$ to both sides of the equation.

$a = 9$

Step 8

Set $a - 8$ equal to $0$ and solve for $a$ .

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Step 8.1

Set $a - 8$ equal to $0$ .

$a - 8 = 0$

Step 8.2

Add $8$ to both sides of the equation.

$a = 8$

Step 9

The final solution is all the values that make $(a - 9) (a - 8) = 0$ true.

$a = 9, 8$

Step 10

Exclude the solutions that do not make $\frac{a^{2} + 13}{a - 8} = \frac{17 a - 59}{a - 8}$ true.

$a = 9$