Factorise monic and non-monic quadratic trinomials

Which product has both the required middle term and constant?

These formal algebra tiles show the collected coefficients when two monic linear expressions are multiplied. Filled tiles are positive and hollow tiles negative; they do not represent physical lengths. Change both constants so the middle coefficient is 1 and the constant is −6.

Which pair gives middle coefficient 1 and constant −6?

Two conditions, one factorisation

The two cross-products combine to make the middle term. Multiplication is commutative, so the brackets can swap

1. 1Monic means leading coefficient 1 In the expansion, m + n is the middle coefficient and mn is the constant. Match both to the original trinomial \(\displaystyle \begin{gathered}(x+m)(x+n)\\=x^2+(m+n)x+mn\end{gathered}\)
2. 2Use both conditions The sign and size of the middle coefficient and the constant must both match
3. 3Retain a common factor If every coefficient has a common factor, remove it first and retain it outside the final brackets
4. 4Multiply back Expand all four products. Collecting like terms must restore the original polynomial

What product must the two constants have?

What sum must the two constants have?

The constants have product 12 and sum 7. Enter the smaller constant

Target

\(\displaystyle x^2+7x+12\)

Expression

\(\displaystyle (x+3)(x+?)\)

Enter the other constant

Which expansion checks every term?

A monic trinomial has constant 15 and middle coefficient −8. What signs must the constants in its two monic linear factors have?

Find constants with product 15 and sum −8. Enter the smaller signed constant

Which product restores both signs?

What product must the constants have?

The constants have sum 1 and product −6. What is the positive constant?

Which repeated factor gives the complete trinomial?

What is the highest positive numerical common factor?

Which complete factorisation reproduces the expression and retains the common factor?

Can x² + x + 1 be written as two monic linear brackets with integer constants?

Extension: a leading coefficient other than 1

Monic quadratic factorisation is Year 9 core. Non-monic examples extend the same reasoning. For ax² + bx + c, split the middle term using integers with product ac and sum b, then group

1. 1Compute ac For 6x² + 7x −3, the required product is 6 × (−3) = −18.
2. 2Find the split −2 and 9 have product −18 and sum 7.
3. 3Group with matching brackets 6x² −2x + 9x −3 becomes 2x(3x −1) + 3(3x −1)
4. 4Check the product (2x + 3)(3x −1) expands back to 6x² + 7x −3.

What is the required product ac?

Split the middle coefficient 7 into two integers with product −18. Enter the smaller integer

Target

\(\displaystyle 6x^2+7x-3\)

Expression

\(\displaystyle 6x^2-2x+\Box x-3\)

What is the other split coefficient?

Which product expands to the original trinomial?