Quadratics and inequalities · Extension
導入済み| English | 意味 |
|---|---|
| discriminant/dɪˈskrɪmɪnənt/ | 判別式 |
Which widths make enough space?
- A rectangular enclosure has area x(10-x). What widths give at least 21 square metres?
- This lesson studies discriminant 判别式: The quantity b²-4ac that determines the real roots of ax²+bx+c=0.
The method
- Factor where possible; otherwise complete the square or use the quadratic formula.
- A quadratic inequality needs the sign on intervals, not only the roots.
- The discriminant identifies repeated or missing real roots.
The discriminant of x²-6x+9=0 is 0. What does that tell you?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
b²-4ac=36-36=0 means the parabola just touches the axis: one repeated root, here x=3. No real roots needs a negative discriminant.
A worked case
The condition is x(10-x)≥21, so x²-10x+21≤0. Factor (x-3)(x-7)≤0. The upward parabola is nonpositive between its roots, giving 3≤x≤7. The maximum area is 25 at x=5.
Quadratics and inequalities
y=(x²-10x+21)/4
The graph rescales the vertical axis by a positive factor 1/4. Check why its roots remain 3 and 7.
Find the smaller root of x²-10x+21=0.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Factorise to (x-3)(x-7)=0. The smaller root is 3.
Where it goes wrong
- Multiplying an inequality by a negative number reverses its direction.
- A sketch must show which side of each root satisfies the inequality.
- Geometry may restrict x further.
Not true: A positive discriminant means that a quadratic has no real roots. A positive discriminant means two distinct real roots; a negative discriminant means no real roots.
Find the discriminant of x²-6x+9=0.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
The discriminant is (-6)²-4×1×9=0.
A positive discriminant means that a quadratic has no real roots.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
A positive discriminant means two distinct real roots; a negative discriminant means no real roots.
Applying it
- Use the vertex to interpret an optimum.
- Check an endpoint and a point between the roots; the algebra and graph should tell the same story.
Find the maximum value of x(10-x).
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Complete the square: x(10-x)=25-(x-5)². The maximum is 25.
Match each discriminant value to the roots of the quadratic.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
The discriminant counts real roots before any solving; it tells you what kind of answer to expect.
In your course
- OxfordAQA International GCSE Mathematics (9260), Extension tier, specification section 3.2.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Discriminant: The quantity b²-4ac that determines the real roots of ax²+bx+c=0.
- Factor where possible; otherwise complete the square or use the quadratic formula.
- Watch for: multiplying an inequality by a negative number reverses its direction.
Write from the inequality 2 less than or equal to x less than or equal to 9
- Solve (x - 2)(x - 9) less than or equal to 0.
- For an upward parabola the solution is between the roots.
Solve (x - 2)(x - 9) less than or equal to 0.
A positive product sits outside the roots. A negative or zero product sits between them, including the roots.
Which values satisfy the inequality?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
The graph is on or below the axis only between the roots 2 and 9.