What is a parameter?
In x² = 9 the number 9 is fixed. In x² = a the letter a is a parameter: one fixed number that we have not chosen yet. The unknown we solve for is still x.
So the answer is not one number. The answer says how the solutions depend on a. For example: "if a < 0 there is no solution; if a = 0 there is one; if a > 0 there are two".
The graph method: slide the line y = a
Graphical method in four moves:
- Move everything so the equation reads f(x) = a (a alone on one side).
- Draw the curve y = f(x) once. It does not depend on a.
- Draw the horizontal line y = a. Changing a slides it up or down.
- Count the meeting points for each position. Write the ranges of a where the count stays the same, and the special values where it changes.
Example: |x² − 4| = a. The curve is W-shaped with low points at (−2, 0), (2, 0) and a peak at (0, 4).
- a < 0: 0 solutions
- a = 0: 2 solutions (x = ±2)
- 0 < a < 4: 4 solutions
- a = 4: 3 solutions
- a > 4: 2 solutions
When a sits inside the formula: discriminant
For x² − 4x + a = 0 the number of roots depends on the discriminant D = b² − 4ac = 16 − 4a.
- D > 0, that is a < 4: two roots.
- D = 0, that is a = 4: one root.
- D < 0, that is a > 4: no real root.
For "x² + 2ax + 4 > 0 for every x" we need no root and the parabola above the axis: D = 4a² − 16 < 0, so −2 < a < 2.
You can also draw it: x² − 4x = −a is the parabola with lowest point −4 and a line at height −a.
Inequalities and cases
In ax > 2 the sign of a decides the answer, so split into cases:
- a > 0: divide, keep the sign: x > 2/a.
- a < 0: divide, flip the sign: x < 2/a.
- a = 0: it reads 0 > 2, false: no solution.
Always say what happens at a = 0 and at the special values you found. A full answer lists every case.
Try it: predict, then check
Open the 3D board on |x − 1| + |x + 1| = a. Before you slide, guess how many solutions a = 1 gives. Then slide to a = 1 and read it. Try a = 2 and a = 3. At home: fill a bowl with water and look at the water surface touching the wall. A deep bowl touches in two places; a bowl with a flat bottom can touch along a whole line. That is the same picture.
Key formulas and definitions
- f(x) = a: solutions = meeting points of y = f(x) and y = a
- D = b² − 4ac: D > 0 two roots, D = 0 one root, D < 0 none
- x² + bx + c > 0 for all x ⇔ D < 0
- ax > c: a > 0 → x > c/a; a < 0 → x < c/a; a = 0 → check 0 > c
Worked examples
1. How many solutions does x² = a have?
Draw y = x² and the line y = a. If a < 0 the line is below the curve: 0 solutions. If a = 0 it touches at the bottom: 1 solution. If a > 0 it cuts twice: 2 solutions (x = ±√a).
2. For which a does x² − 4x + a = 0 have two different roots?
D = (−4)² − 4·1·a = 16 − 4a. Two different roots need D > 0, so 16 − 4a > 0, so a < 4.
3. Discuss |x| = a.
The curve y = |x| is a V with the point at (0, 0). a < 0: none. a = 0: one (x = 0). a > 0: two (x = ±a).
4. How many solutions does |x² − 4| = a have when a = 2?
x² − 4 = 2 gives x² = 6, x = ±√6. x² − 4 = −2 gives x² = 2, x = ±√2. Four different solutions.
5. Find the number of solutions of x² − 2x = a for each a.
x² − 2x = (x − 1)² − 1, a parabola with lowest point (1, −1). a < −1: none. a = −1: one (x = 1). a > −1: two.
6. Solve ax > 2 for all values of a.
a > 0: x > 2/a. a < 0: dividing by a flips the sign, x < 2/a. a = 0: 0 > 2 is false, so no solution.
7. For which a is x² + 2ax + 4 > 0 for every x?
The parabola opens up. It stays above the axis if it has no real root: D = (2a)² − 4·4 = 4a² − 16 < 0, so a² < 4, so −2 < a < 2.
8. For which a does |x − 1| + |x + 1| = a have exactly two solutions?
The curve is flat at height 2 between x = −1 and 1, then climbs with slope 2 on both sides. a < 2: none. a = 2: infinitely many. a > 2: two solutions x = ±a/2. So a > 2.
Common mistakes
- Giving one number as the answer. For a parameter problem the answer is a list of cases for a.
- Forgetting the special values, like the a where the line touches the curve only once.
- Dividing by a without checking a = 0 and the sign of a.
- Counting a touching point twice. A touch is one solution (a double root), not two.