The arc of a thrown ball
Toss a basketball and its path curves up then down — a parabola, described by an x² equation. Factoring that quadratic tells you exactly where the ball leaves and returns to hoop height, the two spots the curve crosses the line.
When x is squared
A quadratic equation has the form ax² + bx + c = 0. Many can be solved by factoring into two binomials, then applying the zero-product property: if two things multiply to zero, at least one of them must be zero.
Zero-product property
If (x − p)(x − q) = 0, then x = p or x = q. Factoring turns one hard equation into two easy ones.
x² − 5x + 6 = 0 factors to (x − 2)(x − 3) = 0, so x = 2 or x = 3.
Lab · Factor and solve
- Take x² − 7x + 12 = 0.
- Find two numbers that multiply to 12 and add to −7.
- Write the factors and read off the solutions.
What you should see: −3 and −4 multiply to 12 and add to −7, so (x − 3)(x − 4) = 0 and x = 3 or x = 4.