Completing the square and quadratic structure
| English | Português |
|---|---|
| completed square/kəmˈpliːtɪd skweə/ | completed square |
A quadratic model reaches a lowest value. Which form tells us where it happens without drawing every point?
- A quadratic model reaches a lowest value. Which form tells us where it happens without drawing every point?
- This lesson studies completed square 配方形式: A quadratic written as a multiple of a squared bracket plus a constant.
Choose the mathematical structure
- For x²+bx, add and subtract (b/2)² to obtain (x+b/2)²−(b/2)². If the leading coefficient a is not 1, factor a from the x² and x terms first. The form a(x−h)²+k gives vertex (h,k), axis x=h, and a minimum k if a>0 or a maximum k if a<0.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines completed square?
A quadratic written as a multiple of a squared bracket plus a constant.
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.
2x²−12x+11=2(x²−6x)+11=2[(x−3)²−9]+11=2(x−3)²−7. Its minimum is −7 at x=3. To solve 2x²−12x+11=0, obtain (x−3)²=7/2, so x=3±√(7/2). The roots lie equally far from x=3. In general ax²+bx+c=0 becomes (x+b/(2a))²=(b²−4ac)/(4a²), for a≠0, leading to x=(−b±√(b²−4ac))/(2a). A negative discriminant means no real roots; zero gives one repeated root.
Completing the square and quadratic structure
For x²+bx, add and subtract (b/2)² to obtain (x+b/2)²−(b/2)²
Find the first valid or invalid step in a quadratic method.
Find the x-coordinate of the minimum of 2x²−12x+11.
The vertex is x=−b/(2a)=12/4=3.
Test a tempting shortcut
- Half the coefficient inside the factored bracket, not the original coefficient of x. Keep the outside multiplier when subtracting the added square. A positive squared bracket does not imply the whole quadratic is positive: its constant can be negative.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A completed-square expression always represents a quadratic with a minimum. This claim is false. Explain which definition or assumption it violates.
Find that minimum value.
At x=3, y=18−36+11=−7.
A completed-square expression always represents a quadratic with a minimum.
Half the coefficient inside the factored bracket, not the original coefficient of x. Keep the outside multiplier when subtracting the added square. A positive squared bracket does not imply the whole quadratic is positive: its constant can be negative.
Interpret a new situation
- For the same model, y≤1 becomes 2(x−3)²≤8, giving 1≤x≤5. Check both endpoints. A contextual restriction such as x≥0 must be combined with this interval. For y=−2(x−3)²+7, the same vertex x-coordinate gives a maximum instead of a minimum.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the larger solution of 2x²−12x+11=11.
Set y=11: 2x(x−6)=0, so the larger root is 6.
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
- 7357 · A-level · B. 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.
A quadratic written as a multiple of a squared bracket plus a constant. Choose the relationship, show the method, check its assumptions and interpret the result.