Factorising and solving simple quadratics · Core
הוצג| English | משמעות |
|---|---|
| discriminant/dɪˈskrɪmɪnənt/ | מבחן |
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
- Expand brackets and factorise simple quadratics.
- Solve by setting each factor equal to zero, and use a graph to interpret the roots.
The equation x²-10x+21=0 factorises as (x-3)(x-7)=0. What are its roots?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
Each factor equal to zero gives one root: x-3=0 or x-7=0. Setting the numbers themselves to zero confuses coefficients with roots.
A worked case
x²-10x+21=(x-3)(x-7). Hence the equation x²-10x+21=0 has roots 3 and 7. Check each root by substitution and mark both intercepts on the graph.
Factorising and solving simple quadratics
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
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 larger root of x²-10x+21=0.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
The factorised roots are 3 and 7; the larger is 7.
A positive discriminant means that a quadratic has no real roots.
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
A positive discriminant means two distinct real roots; a negative discriminant means no real roots.
Applying it
- This Foundation/Core lesson uses factorisation and graphical roots; the discriminant, quadratic formula and quadratic inequalities are reserved for the advanced tier.
Find (5-3)(5-7).
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Substitute 5: (5-3)(5-7)=2×(-2)=-4.
Match each step to its purpose in solving by factorisation.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
A product is zero only when one of its factors is zero, so each factor gives one solution.
In your course
- OxfordAQA International GCSE Mathematics (9260), Core 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.
- Expand brackets and factorise simple quadratics.
- Watch for: multiplying an inequality by a negative number reverses its direction.
Write from the roots 2 and 6
- Solve x squared minus 8x plus 12 = 0 by factorising.
- The roots are not the coefficients 8 and 12.
Solve x squared minus 8x plus 12 = 0. What are the roots?
2 plus 6 is 8 and 2 times 6 is 12, which matches the quadratic.
What are the roots?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
The factors give x = 2 and x = 6. Other options — 3 and 7 (the lesson roots of x squared minus 10x plus 21); 8 and 12 (the coefficients).