Factorising and solving simple quadratics · Foundation
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| root/ruːt/ |
Which widths make enough space?
- A rectangular enclosure has area x(10-x). What widths give at least 21 square metres?
- This lesson studies root 零点: An input for which the expression has value zero.
Choose the mathematical structure
- Expand brackets and factorise simple quadratics. Solve by setting each factor equal to zero, and use a graph to interpret the roots.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines root?
An input for which the expression has value zero.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
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.
Factorise to (x-3)(x-7)=0. The smaller root is 3.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A positive discriminant means that a quadratic has no real roots. This claim is false. Explain which definition or assumption it violates.
Find the larger root of x²-10x+21=0.
The factorised roots are 3 and 7; the larger is 7.
A positive discriminant means that a quadratic has no real roots.
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.
Interpret a new situation
- This Foundation/Core lesson uses factorisation and graphical roots; the discriminant, quadratic formula and quadratic inequalities are reserved for the advanced tier.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find (5-3)(5-7).
Substitute 5: (5-3)(5-7)=2×(-2)=-4.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Foundation · 3.2. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
An input for which the expression has value zero. Choose the relationship, show the method, check its assumptions and interpret the result.