Factoring quadratic expressions

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 the constant term.

Factoring expressions Remember: factoring an expression is the opposite of multiplying it. Multiplying Factoring (a + 1)(a + 2) a2 + 3a + 2 Often: When we multiply an expression we remove the parentheses. When we factor an expression we write it with parentheses.

Factoring quadratic expressions Quadratic expressions of the form x2 + bx + c can be factored if they can be written using parentheses as (x + d)(x + e) where d and e are integers. If we multiply (x + d)(x + e) we have: (x + d)(x + e) = x2 + dx + ex + de = x2 + (d + e)x + de Students will require lots of practice to factor quadratics effectively. This slide explains why when we factor 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.

Factoring quadratic expressions 1 Factor 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. The lesser 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 factored form.