Non-Linear Equations

To factor a quadratic expression means to write it as the product of two linear expressions.

To factor x² + bx + c, find the numbers p and q such that p + q = b and pq = c.

Example:

x² +7x + 10 = (x + 5)(x + 2)

5 + 2 = 7 and 5 × 2 = 10

To factor ax² + bx + c where a ≠ 1, find the factors of a (m and n) and c ( p and q) so that the sum of the outer and inner products (mq and pn) is b.

ax² + bx + c = (mx + p)(nx + q)

Example:

x² + 4x - 12 matches (x + 6)(x - 2)

“The factor pairs for 12 are 12 and 1, 6 and 2, and 3 and 4. Because the last term in the quadratic expression is negative, one factor is positive and one is negative. These factors must have a sum of 4, so the factored expression is (x + 6)(x - 2).”

A quadratic function has a U-shaped graph called a parabola. A quadratic function can be written in the form ax² + c, where a is called the leading coefficient. If a is positive, the parabola opens up. If a is negative, the parabola opens down. If the absolute value of a is greater than 1, then the parabola will be narrower than y = x². If the absolute value of a is less than 1, then the parabola will be wider than y = x².

The vertex of a parabola is the point from which the graph opens up or down.

Adding values to and subtracting values from ax² translates the vertex along the y-axis.

In the form y = (x - a)(x - b), the vertex is located at (a, b) .

matches .

“The vertex is at (0, 4). Because the leading coefficient is positive, the graph opens up. Because the leading coefficient is less than 1, the graph will be wider than y = x².”