Quadratic expressions A quadratic expression is an expression in which the highest power of the variable is 2. For example, t2 2 x2 – 2, w2 + 3w + 1, 4 – 5g2 , The general form of a quadratic expression in x is: ax2 + bx + c (where a = 0) x is a variable. As well as the highest power being two, no power in a quadratic expression can be negative or fractional. Compare each of the quadratic expressions given with the general form. In x2 – 2, a = 1, b = 0 and c = –2. In w2 + 3w + 1, a = 1, b = 3 and c = 1. This is a quadratic in w. In 4 – 5g2, a = –5, b = 0 and c = 4. This is a quadratic in g. In t2/2, a = ½, b = 0 and c = 0. This is a quadratic in t. a is a fixed number and is the coefficient of x2. b is a fixed number and is the coefficient of x. c is a fixed number and is a constant term.

Factorizing expressions Remember: factorizing an expression is the opposite of expanding it. Expanding or multiplying out Factorizing (a + 1)(a + 2) a2 + 3a + 2 Often: When we expand an expression we remove the brackets. When we factorize an expression we write it with brackets.

Factorizing quadratic expressions Quadratic expressions of the form x2 + bx + c can be factorized if they can be written using brackets as (x + d)(x + e) where d and e are integers. If we expand (x + d)(x + e) we have, (x + d)(x + e) = x2 + dx + ex + de = x2 + (d + e)x + de Pupils will require lots of practice to factorize quadratics effectively. This slide explains why when we factorize an expression in the form x2 + bx + c to the form (x + d)(x + e) the values of d and e must be chosen so that d + e = b and de = c. (x + d)(x + e) = x2 + (d + e)x + de is an identity. This means that the coefficients and constant on the left-hand side are equal to the coefficients and constant on the right-hand side. Comparing this to x2 + bx + c we can see that: The sum of d and e must be equal to b, the coefficient of x. The product of d and e must be equal to c, the constant term.

Factorizing quadratic expressions 1 Factorize the given expression by finding two integers that add together to give the coefficient of x and multiply together to give the constant. It may be a good idea to practice adding and multiplying negative numbers before attempting this activity. Use slide 31 in N1.2 Calculating with integers to do this if required. The lower of the two hidden integers will be given first in each case.

Matching quadratic expressions 1 Select a quadratic expression and ask a volunteer to find its corresponding factorization.

Factorizing quadratic expressions Quadratic expressions of the form ax2 + bx + c can be factorized if they can be written using brackets as (dx + e)(fx + g) where d, e, f and g are integers. If we expand (dx + e)(fx + g)we have, (dx + e)(fx + g)= dfx2 + dgx + efx + eg = dfx2 + (dg + ef)x + eg Discuss the factorization of quadratics where the coefficient of x2 is not 1. Most examples at this level will have a as a prime number so that there are only two factors, 1 and the number itself. Comparing this to ax2 + bx + c we can see that we must choose d, e, f and g such that: a = df, b = (dg + ef) c = eg

Factorizing quadratic expressions 2 Factorize each given expression using trial and improvement and the relationships shown on the previous slide. For each expression in the form ax2 + bx + c, start by using the pen tool to write down pairs of integers that multiply together to make a and pairs of integers that multiply together to make c. Use these to complete the factorization.

Matching quadratic expressions 2 Select a quadratic expression and ask a volunteer to find its corresponding factorization.

Factorizing the difference between two squares A quadratic expression in the form x2 – a2 is called the difference between two squares. The difference between two squares can be factorized as follows: x2 – a2 = (x + a)(x – a) For example, See slide 40 to demonstrate the expansion of expressions of the form (x + a)(x – a). Pupils should be encouraged to spot the difference between two squares whenever possible. 9x2 – 16 = (3x + 4)(3x – 4) 25a2 – 1 = (5a + 1)(5a – 1) m4 – 49n2 = (m2 + 7n)(m2 – 7n)

Factorizing the difference between two squares

Matching the difference between two squares Select an expression involving the difference between two squares and ask a volunteer to find the corresponding factorization.