Verify quadratic products
Expand and collect like terms to establish an identity; use the selected product to read ordered coefficients
- Compare three coefficients For two linear factors, check the leading product, both cross-products and the constant product
- A square has two cross-products (ax + b)² means (ax + b)(ax + b), not a²x² + b²
- Read the displayed order In the practice coefficient table, factor 1 is the left factor of the correct choice in part(a). Factors may be written in another order elsewhere
- Include signs The constant of x −5 is −5. A squared bracket is one displayed base factor used twice
Worked example
Check all four products
\[(2x+3)(3x-1)\]
- \(\displaystyle (2x)(3x)=6x^2\)Multiply the leading terms
- \(\displaystyle (2x)(-1)=-2x\)First cross-product
- \(\displaystyle 3(3x)=9x\)Second cross-product
- \(\displaystyle 3(-1)=-3\)Multiply the constants
- \(\displaystyle 6x^2+7x-3\)Combine −2x and 9x to get 7x
Common mistakes
Checking only the constant
✕\(\displaystyle \begin{gathered}x^2+5x+6\\=(x+1)(x+6)\end{gathered}\)✓\(\displaystyle \begin{gathered}x^2+5x+6\\=(x+2)(x+3)\end{gathered}\)
Both products have constant 6, but only the second has middle coefficient 5.Omitting the cross-products
✕\(\displaystyle \begin{gathered}(x+3)^2\\=x^2+9\end{gathered}\)✓\(\displaystyle \begin{gathered}(x+3)^2\\=x^2+6x+9\end{gathered}\)
A square is a product of two identical bracketsDifficulty levels
standardVerify monic products, quadratic products with leading coefficients, and repeated factors
Questions about this topic
Why can factor order differ between the two topics?+
Multiplication allows the brackets to swap. The dedicated factor-entry task asks for a canonical order; this verification task explicitly follows the correct displayed choice
How do I read a squared factor?+
Read the coefficients inside its displayed base bracket. The exponent means that bracket is multiplied by itself
Are all these forms Year 9 core?+
Monic factorisation is Year 9 core. Non-monic products extend the same distributive reasoning
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 8: Algebraic expressions and fractions
In this chapter
Expand binomial and special products8A Expanding and special products: Check equivalent expanded formsSubstitute values into a formulaFactorise by common factors and difference of perfect squares8B Common factors and difference of squares: Verify common-factor and square factorisationsFactorise four-term expressions by groupingFactorise monic and non-monic quadratic trinomials8D Quadratic factorisation: Verify quadratic factorisationsSimplify and operate on algebraic fractionsSolve equations that contain algebraic fractions