Sum and product of roots, hidden quadratics, line meets parabola
Read the sum and product of the roots from the coefficients, solve an x to the fourth equation by a change of letter, and find where a line meets a parabola.
- UK: Year 11
- France: grade 11
- Germany: grade 9
- Iran: grade 11
- Syria: grade 10
- Palestine: grade 11
- Ukraine: grade 8
- Russia: grade 8
Guide
If x2 + bx + c = 0 has roots r and s, then (x − r)(x − s) = x2 − (r + s)x + rs. So the roots add to r + s = −b and multiply to rs = c. For ax2 + bx + c = 0: sum −b ÷ a, product c ÷ a. You can read these without finding the roots.
Some equations are not quadratics but turn into one. In x4 − 5x2 + 4 = 0 let u = x2; then u2 − 5u + 4 = 0. Solve for u first, then use x2 = u to find x.
A line and a parabola meet where their y values are equal. Set the two expressions for y equal; you get a quadratic, and each root of it is the x of a meeting point.
In each task, mark the roots or x values with points on the number line.