Simplify before naming
An equation's visible form may mislead. Expand both sides, move every term to one side and collect like terms before deciding its degree.
Learning goals
- Recognise a quadratic equation after full simplification.
- Write standard form and identify signed coefficients.
- Translate a situation into a quadratic model.
Concept lesson
Expand and collect
Use (x+a)^2=x^2+2ax+a^2 and distribute every product before moving terms and collecting equal powers.
Quadratic equation test
A quadratic equation reduces to ax^2+bx+c=0 where a, b, c are real and a is not zero. b or c may be zero; a may not.
Cancellation changes the apparent degree
For (x+2)^3=x^3+4, expand the left. The x^3 terms cancel, leaving 6x^2+12x+4=0, which is quadratic.
Standardise and label signed coefficients
(2x+3)x=x^2+1 gives 2x^2+3x=x^2+1, hence x^2+3x-1=0. Therefore a=1, b=3, c=-1.
Form from consecutive integers
Let the smaller positive integer be x; the next is x+1. Product 306 gives x(x+1)=306, so x^2+x-306=0.
Is a really non-zero?
Classify kx^2+3x-2=0 for k=0 and k=4. Explain why the same template changes type.
Translate and reread
A rectangle's length is one metre more than twice its breadth and area is 528 square metres. Define breadth x, form the equation, then reread every term in context.
Diagnose a false classification
A student classifies (x-2)(x+1)=(x-1)(x+3) as quadratic by looking at products. Expand both sides and correct the conclusion.
Standard-form marks
Show expansion, move all terms to one side, collect descending powers, write =0 and identify coefficients with their signs when requested.
Exit ticket
Simplify and classify x(x+2)=6; identify a,b,c. Then form the quadratic equation for two consecutive positive integers whose product is 72.