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 5Omitting 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
standardMonic, simple non-monic and repeated binomial factors, with ordered coefficient data
examVerify the same factor patterns with strategy, coefficient and exact substitution checks
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
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 10: Quadratic and non-linear graphs
In this chapter
Expand binomial products and special products10A Expanding and special products: Verify binomial and special-product expansions10B Common factors and difference of squares: Factorise common factors and differences of squares10B Common factors and difference of squares: Verify common-factor and square-difference products10C Quadratic factorisation and fractions: Factorise a quadratic expression completelySimplify an algebraic fraction by factorising and cancelling10C Quadratic factorisation and fractions: Verify monic and simple non-monic factorisationsVerify quadratic fraction cancellations and exact valuesSolve a quadratic equation by taking square rootsSolve a quadratic equation using the null factor lawSet up and solve a quadratic application problemIdentify and interpret parabola features10G Parabola features and transformations: Distinguish linear and quadratic relationsRead and use circle equations and graphsRead and use exponential growth and decay graphs10H Circles and exponential graphs: Classify a relation as linear, quadratic, circle or exponentialConnect reciprocal rules, tables and graphs