Precalculus Examples

$f (x) = \frac{a x^{2} + b x + c}{d x^{2} + e x + f}$

Step 1

Subtract $\frac{a x^{2} + b x + c}{d x^{2} + e x + f}$ from both sides of the equation.

$y - \frac{a x^{2} + b x + c}{d x^{2} + e x + f} = 0$

Step 2

To write $y$ as a fraction with a common denominator, multiply by $\frac{d x^{2} + e x + f}{d x^{2} + e x + f}$ .

$\frac{y (d x^{2} + e x + f)}{d x^{2} + e x + f} - \frac{a x^{2} + b x + c}{d x^{2} + e x + f} = 0$

Step 3

Combine the numerators over the common denominator.

$\frac{y (d x^{2} + e x + f) - (a x^{2} + b x + c)}{d x^{2} + e x + f} = 0$

Step 4

Simplify the numerator.

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Apply the distributive property.

$\frac{y (d x^{2}) + y (e x) + y f - (a x^{2} + b x + c)}{d x^{2} + e x + f} = 0$

Rewrite using the commutative property of multiplication.

$\frac{y d x^{2} + e y x + y f - (a x^{2} + b x + c)}{d x^{2} + e x + f} = 0$

Apply the distributive property.

$\frac{y d x^{2} + e y x + y f - (a x^{2}) - (b x) - c}{d x^{2} + e x + f} = 0$

Simplify.

$\frac{y d x^{2} + e y x + y f - a x^{2} - b x - c}{d x^{2} + e x + f} = 0$