ISM: Introductory Mathematical Analysis
Section 1.2 (u − 4)(u − 9) = 0 u–4=0 u=4
b. To find the year in which the number of teachers is 252,041 we solve the equation 2
175.15t + 3641.68t + 26032 = 252041 2 175.15t + 3641.68t − 226009 = 0
8. 2( z 2 + 4 z + 4) = 0 2( z + 2) 2 = 0 z+2=0 z = −2
Applying the quadratic formula with a = 175.15, b = 3641.68 and c = −226009 gives −3641.68 ± (3641.68)2 − 4(175.15)(−226009) 2(175.15) −3641.68 ± 13099.76 = ≈ −47.7, 27.0 350.30
t=
As t is the number of years after 1990, the positive solution 27 is the reasonable one, so the number of female secondary-school teachers will be 252,041 in 2017. Problems 1.2 2
1. x − 4 x + 4 = 0 ( x − 2)2 = 0 x–2=0 x=2
2. (t + 1)(t + 2) = 0 t+1=0 t = –1 3.
or t + 2 = 0 or t = –2
2
t − 6t + 8 = 0 (t − 4)(t − 2) = 0 t − 4 = 0 or t − 2 = 0 t = 4 or t=2
4. (x – 2)(x + 5) = 0 x–2=0 x=2
or x + 5 = 0 or x = –5
5. x 2 − 2 x − 3 = 0 (x – 3)(x + 1) = 0 x–3=0 x=3
or x + 1 = 0 or x = –1
6. (x – 4)(x + 4) = 0 x–4=0 x=4 7. u 2 − 13u = −36 2
u − 13u + 36 = 0
or u – 9 = 0 or u = 9
9. x 2 − 4 = 0 (x – 2)(x + 2) = 0 x–2=0 x=2
or x + 2 = 0 or x = –2
10. 3u (u − 2) = 0 u=0 u=0
or u − 2 = 0 or u = 2
11. t 2 − 5t = 0 t (t − 5) = 0 t=0 t=0
or t – 5 = 0 or t = 5
12. x 2 + 9 x + 14 = 0 (x + 7)(x + 2) = 0 x+7=0 x = –7
or x + 2 = 0 or x = –2
13.
9 x 2 + 4 = −12 x 9 x 2 + 12 x + 4 = 0 (3x + 2)2 = 0 3x + 2 = 0 3 x = −2 2 x=− 3
14. 2 z 2 + 9 z − 5 = 0 (2z – 1)(z + 5) = 0 2z – 1 = 0 or z + 5 = 0 1 z= or z = –5 2 15. v(3v − 5) = −2
or x + 4 = 0 or x = –4
3v 2 − 5v = −2 3v 2 − 5v + 2 = 0 (3v − 2)(v − 1) = 0 3v – 2 = 0 2 v= 3
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or v − 1 = 0 or v = 1