Quadratic trinomials and factorisation

To expand the brackets, multiply each term in the first pair of brackets by each term in the second:

(x − a)(x − b) = x² − bx − ax + ab.

Combine the like terms:

(x − a)(x − b) = x² − (a + b)x + ab.

1. Multiply x · x.
2. Combine the two terms containing x: −bx − ax.
3. Multiply the constant terms: (−a)(−b) = ab.

Pay close attention to the signs. If a negative number appears in brackets, for example x − (−3), the expression is equal to x + 3.

For ax² + bx + c = 0, the discriminant is D = b² − 4ac. D > 0 gives two real roots, D = 0 one and D < 0 none. For D ≥ 0, x = (−b ± √D)/(2a). In the given equation the x² coefficient is 1.

Find m and n whose sum is the x coefficient and whose product is the constant. Then x² + (m + n)x + mn = (x + m)(x + n). Check by expanding.

A root x₁ gives the factor x − x₁ because substituting x = x₁ makes it zero. The same holds for the second root. Thus a monic quadratic factors as (x − x₁)(x − x₂); a negative root gives a plus sign in its factor.

Use the perfect-square identity:

(x + a)² = x² + 2ax + a².

Check that the final term is the square of a and that the coefficient of x equals 2a. When both conditions hold, the trinomial can be written as (x + a)².

You can also find a by dividing the coefficient of x by 2.

Find m and n whose sum is the x coefficient and whose product is the constant. Then x² + (m + n)x + mn = (x + m)(x + n). Check by expanding.

To expand the brackets, multiply each term in the first pair of brackets by each term in the second:

(x − a)(x − b) = x² − bx − ax + ab.

Combine the like terms:

(x − a)(x − b) = x² − (a + b)x + ab.

1. Multiply x · x.
2. Combine the two terms containing x: −bx − ax.
3. Multiply the constant terms: (−a)(−b) = ab.

Pay close attention to the signs. If a negative number appears in brackets, for example x − (−3), the expression is equal to x + 3.

Use the perfect-square identity:

(x + a)² = x² + 2ax + a².

Compare it with the given trinomial x² + kx + c:

• the constant term c must equal a²;
• the middle term kx must match 2ax.

Therefore, to find k, multiply a by 2. Substitute the result for k to check that the two expressions are identical.

The vertex form of a parabola is y = a(x − m)² + n.

Its vertex coordinates can be read directly:

V(m, n).

Pay close attention to the sign inside the brackets. For example, if the expression is x − (−3), the x-coordinate of the vertex is −3, not 3.

The quadratic trinomial x² − sx + p can be factorised into two factors:

(x − a)(x − b).

Find two numbers a and b that satisfy both conditions:

• their sum equals s: a + b = s;
• their product equals p: a · b = p.

First list the factor pairs of p, then select the pair whose sum equals s.

You can check the result by expanding the brackets: (x − a)(x − b) = x² − (a + b)x + ab.