Mathematics · 14 / 23
Quadratics: discriminant and roots
High school · intermediate · Worked example and 4 exercises with solutions, at your pace.
The principle, simply
A polynomial ax² + bx + c with nonzero a draws a parabola. Its zeros are intersections with the horizontal axis. The discriminant Δ = b² − 4ac determines real roots: two when positive, one repeated root when zero, none when negative. Factoring then helps analyse signs.
- 01Identify coefficients
- 02Calculate Δ
- 03Choose the case
- 04Check roots
WORKED EXAMPLE
From principle to reasoning
For x² − 3x + 2, Δ = 9 − 8 = 1. Roots are (3 − 1)/2 = 1 and (3 + 1)/2 = 2, giving (x − 1)(x − 2).
From roots to signs
For (x − 1)(x − 2), mark 1 and 2. Before 1 both factors are negative, giving a positive product. Between them signs differ, giving a negative product. After 2 both are positive. Roots give zero. A negative leading coefficient reverses signs.
Solve a problem in this subject
- Define the objects
State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.
- Choose a transformation
Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.
- Show the reasoning
Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.
- Test and conclude
Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.
YOUR TURN
01 · Try before checking
Solve x² − 4 = 0.
One hint
Factor a difference of squares.
Reveal and understand the solution
(x − 2)(x + 2) = 0, so x = 2 or −2. The discriminant 16 confirms two roots.
Compare method, conditions and answer; identify the exact part to revisit.
IMPROVE THROUGH PRACTICE
Three additional challenges with solutions
Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.
Solve x² − 5x + 6 = 0 by factoring.
A hint to get started
Find two numbers with product 6 and sum 5.
Step-by-step solution
- x² − 5x + 6 = (x − 2)(x − 3).
- A zero product has a zero factor.
- The roots are 2 and 3.
Which step did you miss? Retry without help, then explain why the method works.
How many real roots does 2x² + x + 3 have?
A hint to get started
Find the discriminant before taking a root.
Step-by-step solution
- Δ = 1² − 4 × 2 × 3 = −23.
- Δ < 0 means no real roots.
- The polynomial still has values; it is positive throughout.
Which step did you miss? Retry without help, then explain why the method works.
Area is A(x) = −x² + 8x for 0 ≤ x ≤ 8. Find the maximum.
A hint to get started
Complete the square or find the vertex.
Step-by-step solution
- A(x) = −(x − 4)² + 16.
- The square is smallest at the permitted value x = 4.
- Maximum area: 16 square units.
Which step did you miss? Retry without help, then explain why the method works.
The trap to avoid
If a = 0, use a linear equation rather than the quadratic formula.
Without notes, can you…
- Explain the principle in your own words?
- Rebuild the diagram and its connections?
- Solve the exercise and justify the method?
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