This week in Pre Calculus 11, we reviewed Rational Expressions in preparation for the quiz.

Example question:

 

X² + 3X – 10 ÷ X² – 5X + 6
3x²+13x-10    2x² + 6x – 36

We can turn the division symbol into multiplication by replacing the fraction on the right with its reciprocal.

X² + 3X – 10 .  2x² + 6x – 36
3x²+13x-10    X² – 5X + 6

Now, we factor:

(X + 5) (X – 2) .  (2x+12)(x-3)
(3x-2)(x+5)        (x-2)(x-3)

Now, we can simplify by eliminating common factors.

(X + 5) (X – 2) .  (2x+12)(x-3)
(3x-2)(x+5)        (x-2)(x-3)

This becomes:

(2x+12)
(3x-2).

Now we can further simplify:

2(x+6)
(3x-2)

This can also apply to equations.

1 + 4__      = 2
X   X + 2

Because this is an addition question, we must find a common denominator, we can do that by cross multiplying. Remember that this also applies to the other side of the equation.

 

1(X+2) + 4(X)___  = 2(X+2)(X)
X(X+2)    X(X + 2)       X(X + 2)

Now, we combine the fractions into one.

x + 2 + 4x  = 2X²+ 4x
X² + 2x              X² + 2x

Because both sides of the equation have a common denominator, we can remove the denominator entirely.

5x + 2 = 2x² + 4

From here, we are left with a quadratic equation.

-2x² + 5x + 2 = 4

 

2x² – 5x – 2 = -4

 

2x² – 5x + 2 = 0

 

(2x – 1) (x – 2) = 0

X₁ = ½

X­­­₂ = 2.

In this case, division and multiplication are the same, because solving a division question requires converting it to multiplication before anything else is done. Subtraction in quadratic expressions works the same way as addition, except with subtraction instead of addition.