Factoring Quadratic Expressions By The Difference Of Squares

Slide 2: ax 2+ bx + c when a, b, c are integers b = 0, a, c ≠ 0 a and c have different signs

Slide 3: When there is no term in x , and the coefficient of 2 x and the constant term have different signs, we have the so called difference of squares. 2 2 p −q = p−q pq= pq p−q 2 2 2 2 − p q =q − p =q− pq p=q pq− p Before we factor using the difference of squares, take out the GCF or Greatest Common Factor, if sensible. The GCF is also known as the HCF or Highest Common Factor.

Slide 4: Example 1 2 Factor x −9 The GCF is 1 and there is no point in factoring it out.

Slide 5: 2 2 2 x −9 = x −3 2 x −9 =  x −3 x3 2 Rewrite 9 as 3 ....

Slide 6: 2 2 2 x −9 = x −3 2 x −9 =  x −3 x3 .... and factor using the difference of squares.

Slide 7: Example 2 2 Factor 4 x −25 The GCF is 1 and there is no point in factoring it out.

Slide 8: 2 2 2 4 x −25 = 2 x  −5 2 4 x −25 = 2 x−52 x5 2 2 2 Rewrite 4 x as 2 x  and 25 as 5 ....

Slide 9: 2 2 2 4 x −25 = 2 x  −5 2 4 x −25 = 2 x−52 x5 .... and factor using the difference of squares.

Slide 10: Example 3 2 Factor 18−50 x

Slide 11: 2 2 18−50 x = 29−25 x  2 2 2 18−50 x = 2[3 −5 x  ] 2 18−50 x = 23−5 x 35 x  The GCF is 2 and we factor it out.

Slide 12: 2 2 18−50 x = 29−25 x  2 2 2 18−50 x = 2[3 −5 x  ] 2 18−50 x = 23−5 x 35 x  2 2 2 Rewrite 9 as 3 and 25 x as 5 x  ....

Slide 13: 2 2 18−50 x = 29−25 x  2 2 2 18−50 x = 2[3 −5 x  ] 2 18−50 x = 23−5 x 35 x  .... and factor using the difference of squares. We can reorder in descending powers of x if we wish, but we must factor out −2 in this case. 2 2 18−50 x = −50 x 18 2 = −225 x −9 2 2 = −2[5 x  −3 ] = −25 x−35 x3

Slide 14: Example 4 2 Factor 8 x −25 The GCF is 1 and there is no point in factoring it out.

Slide 15: 8 x −25 =  x  8  −5 2 2 2 8 x −25 =  x  8−5  x  85  2 8 x −25 =  2 x  2−5  2 x  25  2 Rewrite 8 x as  x  8  and 25 as 5 .... 2 2 2 Write  x  8  rather than   8 x  to avoid 2 2 confusion between   8 x  and   8 x  . 2 2

Slide 16: 8 x −25 =  x  8  −5 2 2 2 8 x −25 =  x  8−5  x  85  2 8 x −25 =  2 x  2−5  2 x  25  2 .... and factor using the difference of squares.

Slide 17: 8 x −25 =  x  8  −5 2 2 2 8 x −25 =  x  8−5  x  85  2 8 x −25 =  2 x  2−5  2 x  25  2 We can simplify the irrational number if we wish, x  8=x×  4 × 2 =x ×  4 ×  2=x × 2 × 2 =2 x  2

Slide 18: Example 5 2 Factor 45−8 x The GCF is 1 and there is no point in factoring it out.

Slide 19: 45−8 x =   45  − x  8  2 2 2 45−8 x =   45−x  8   45 x  8  2 45−8 x =  3  5−2 x  2  3  52 x  2  2   45  and 8 x as  x  8  .... 2 2 2 Rewrite 45 as Write  x  8  rather than   8 x  to avoid 2 2 confusion between   8 x  and   8 x  . 2 2

Slide 20: 45−8 x =   45  − x  8  2 2 2 45−8 x =   45−x  8   45 x  8  2 45−8 x =  3  5−2 x  2  3  52 x  2  2 .... and factor using the difference of squares.

Slide 21: 45−8 x =   45  − x  8  2 2 2 45−8 x =   45−x  8   45 x  8  2 45−8 x =  3  5−2 x  2  3  52 x  2  2 We can simplify the irrational numbers if we wish, x  8=x×  4 × 2 =x ×  4 ×  2=x × 2 × 2 =2 x  2  45= 9 × 5 =  9 × 5=3  5

Slide 22: Example 5A 2 Factor 45−8 x This is the same as Example 5. We will take an alternative approach by reordering in descending powers of x and factoring out −1 to 2 make the coefficient of x positive.

Slide 23: 2 2 45−8 x = −8 x 45 45−8 x = − 8 x −45  2 2 [ 45−8 x = −  x  8  −  45  2 2 2 ] 45−8 x = −[  x  8−  45  x  8  45  ] 2 45−8 x = − x  8− 45  x  8  45  2 45−8 x = − 2 x  2−3  5  2 x  23  5  2

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