1
2 Factorizing quadratic expressionsFactorizing an expression is the inverse of expanding it. Expanding or multiplying out Factorizing (x + 1)(x + 2) x2 + 3x + 2 Teacher notes In this example, (x + 1) and (x + 2) are factors of x2 + 3x + 2. When we expand an expression we multiply out the brackets. When we factorize an expression we write it with brackets.
3 Factorizing quadratic expressionsNo constant term Quadratic expressions of the form ax2 + bx can always be factorized by taking out the common factor x. For example: 3x2 – 5x = x(3x – 5) The difference between two squares When a quadratic has no term in x and the other two terms can be written as the difference between two squares, we can use the identity Teacher notes Explain to the students that the approach we take to factorizing a quadratic depends on what form it takes. Later, when the discriminant is introduced, students will be able to check whether a quadratic expression factorizes by checking to see whether b2 – 4ac is a perfect square. a2 – b2 = (a + b)(a – b) to factorize it. For example: 9x2 – 49 = (3x + 7)(3x – 7)
4 Factorizing quadratic expressionsQuadratic expressions with a = 1 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 Teacher notes 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. To check whether quadratic expressions of the form x2 + bx + c factorize, check whether b2 – 4c is a perfect square. This is demonstrated later when the discriminant is introduced. Use the embedded flash activity to demonstrate factorizing quadratics written in this form. In each example the lower of the two hidden integers will be given first.
5 Factorizing quadratic expressionsThe general form Quadratic expressions of the general 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 Teacher notes To check whether quadratic expressions of the form ax2 + bx + c factorize, check whether b2 – 4ac is a perfect square. This is demonstrated later when the discriminant is introduced. Factorizing quadratic expressions where the coefficient of x2 is not 1 requires some practice. Students can start by using the identity shown and using trial and error to find values of d, e, f and g such that df = a, (dg + ef) = b and eg = c.
6 Perfect squares Some quadratic expressions can be written as perfect squares. For example: x2 + 2x + 1 = (x + 1)2 x2 – 2x + 1 = (x – 1)2 x2 + 4x + 4 = (x + 2)2 x2 – 4x + 4 = (x – 2)2 x2 + 6x + 9 = (x + 3)2 x2 – 6x + 9 = (x – 3)2 In general: x2 + 2ax + a2 = (x + a) or x2 – 2ax + a2 = (x – a)2 Teacher notes Encourage students to derive the general form of quadratics that are perfect squares before revealing the perfect square itself. This general form could also be written as x2 + bx + (b/2)2 = (x + b/2)2. How could the quadratic expression x2 + 8x be made into a perfect square? We could add 16 to it.
7 Completing the square Adding 16 to the expression x2 + 8x to make it into a perfect square is called completing the square. x2 + 8x = x2 + 8x + 16 – 16 We can write If we add 16 we then have to subtract 16 so that both sides are still equal. By writing x2 + 8x + 16 we have completed the square and so we can write this as x2 + 8x = (x + 4)2 – 16 In general:
8 Factorizing quadratic expressions 2Teacher notes Factorize each given expression using trial and improvement and the relationships shown on the previous slides. For each expression in the form ax2 + bx + c, you can 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.
9 Want to see more? To see more of what Boardworks can offer, why not order a full presentation, completely free? Head to: This is only a sample of one of hundreds of Boardworks Maths PowerPoints.