2 Quadratic Equations in One Unknown (II)
2 Quadratic Equations in One Unknown (II)
Review Exercise 2 (p. 2.5)
1.
2.
∴ (repeated)
3.
4. Using the quadratic formula,
5. Using the quadratic formula,
∵ is not a real number.
∴ The equation has no real roots.
6. Using the quadratic formula,
7. The x-intercepts of the graph of are –2.0 and 3.0.
Therefore, the roots of are –2.0 and 3.0.
8. The x-intercept of the graph of is 0.5.
Therefore, the root of is 0.5.
9. (a) The graph of intersects the x-axis at two points.
Therefore, the equation has two unequal real roots.
(b) The graph of touches the x-axis at one point.
Therefore, the equation has one double real root.
(c) The graph of does not intersect the x-axis.
Therefore, the equation has no real roots.
10.
∴
11.
∴
12.
∴
13.
∴
To Learn More
To Learn More (p. 2.23)
(a) Mid-point of PQ
(b) Length of PQ
To Learn More (p. 2.30)
(a)
(b)
(c)
(d)
Classwork
Classwork (p. 2.7)
Value of D(D = b2 – 4ac) / Nature of roots
2 unequal real roots / 1 double
real root / No real
roots
(a) / 0 / ü
(b) / 24 / ü
(c) / –31 / ü
(d) / –400 / ü
(e) / 25 / ü
Classwork (p. 2.10)
1. (a) 0 (b) D < 0
2. (a) 2 (b) D > 0
3. (a) 0 (b) D < 0
4. (a) 1 (b) D = 0
5. (a) 1 (b) D = 0
6. (a) 2 (b) D > 0
Classwork (p. 2.18)
Sum of roots / Product of roots(a) / / /
(b) / / /
(c) / / /
(d) / / /
(e) / / /
(f) / / /
Classwork (p. 2.29)
1. (a)
(b)
(c)
(d)
2. (a)
(b)
Classwork (p. 2.31)
Real part / Imaginary part(a) / / 3 / –4
(b) / / –5 / 7
(c) / 8 / 8 / 0
(d) / / 0 /
(e) / / /
Classwork (p. 2.32)
(a) û (b) û (c) ü (d) ü
Quick Practice
Quick Practice 2.1 (p. 2.7)
∵ The equation has one double real root.
∴ D = 0
i.e.
Quick Practice 2.2 (p. 2.8)
∵ The equation has no real roots.
∴
i.e.
∴ The range of values of k is.
Quick Practice 2.3 (p. 2.9)
(a) For the equation,
(b) ∵ is a quadratic equation.
∴ The coefficient ofcannot be zero.
i.e.
(i) ∵ The equationhas two distinct real roots.
∴
∴ The range of values of k isexcept.
(ii) ∵ The equationhas real roots.
∴
∴ The range of values of k isexcept.
Quick Practice 2.4 (p. 2.11)
(a) ∵ The graph oftouches the x-axis at one point P.
∴
i.e.
(b) For m = 9, the corresponding quadratic equation is
∴ The coordinates of P are.
Quick Practice 2.5 (p. 2.11)
(a) ∵ The graph ofhas two
x-intercepts.
∴
i.e.
∴ The range of values of m is.
(b) The smallest integral value of m is 0.
For m = 0, the corresponding quadratic equation is
∴ The x-intercepts of the graph areand 1.
Quick Practice 2.6 (p. 2.15)
(a) The required quadratic equation is:
(b) The required quadratic equation is
Quick Practice 2.7 (p. 2.16)
(a) Sum of roots
Product of roots
∴ The required quadratic equation is
(b) Sum of roots
Product of roots
∴ The required quadratic equation is
Quick Practice 2.8 (p. 2.19)
(a) For the equation ,
∴
(b)
Quick Practice 2.9 (p. 2.20)
Quick Practice 2.10 (p. 2.21)
(a)
(b)
(c)
Quick Practice 2.11 (p. 2.22)
(a)
(b)
Quick Practice 2.12 (p. 2.23)
∵ andare the roots of .
∴
For the required quadratic equation,
∴ The required quadratic equation is
Quick Practice 2.13 (p. 2.29)
(a)
(b)
Quick Practice 2.14 (p. 2.32)
By comparing the real parts, we have
By comparing the imaginary parts, we have
Quick Practice 2.15 (p. 2.33)
(a)
(b)
Quick Practice 2.16 (p. 2.34)
(a)
(b)
(c)
Quick Practice 2.17 (p. 2.35)
(a)
(b)
(c)
Quick Practice 2.18 (p. 2.36)
Using the quadratic formula,
Quick Practice 2.19 (p. 2.37)
∵ is a root of the equation.
∴
∴
From (2),
By substitutinginto (1), we have
Further Practice
Further Practice (p. 2.12)
1.
∵ The equationhas two unequal real roots.
∴
i.e.
∴ The range of values of m is m < 8.
2. (a) ∵ The graph of touches the x-axis at one point P.
∴
i.e.
(b) For k = 5, the corresponding quadratic equation is
∴ The coordinates of P are.
For, the corresponding quadratic equation is
∴ The coordinates of P are.
Further Practice (p. 2.23)
1.
2.
(a)
(b)
Further Practice (p. 2.35)
1.
2. (a)
By comparing the real parts, we have
By comparing the imaginary parts, we have
(b)
By comparing the real parts, we have
By comparing the imaginary parts, we have
……(1)
By substitutinginto (1), we have
Exercise
Exercise 2A (p. 2.12)
Level 1
1. For the equation,
∵
∴ The equation has two unequal real roots.
2. For the equation,
∵
∴ The equation has no real roots.
3. For the equation,
∵
∴ The equation has one double real root.
4. Consider.
∴ The graph ofhas one x-intercept.
5. Consider.
∴ The graph ofhas no x-intercepts.
6. Consider.
∴ The graph ofhas two x-intercepts.
7. ∵ The equationhas one double real root.
∴
i.e.
8. ∵ The equationhas two equal real roots.
∴
i.e.
9. ∵ The equationhas two unequal real roots.
∴
i.e.
∴ The range of values of m is.
10. ∵ The equationhas two unequal real roots.
∴
i.e.
∴ The range of values of m is.
11. ∵ The equationhas no real roots.
∴
i.e.
∴ The range of values of k is.
12. ∵ The equationhas no real roots.
∴
i.e.
∴ The range of values of k is k > 5.
13. ∵ The graph of has only one x-intercept.
∴
i.e.
14. ∵ The graph of has no x-intercepts.
∴
i.e.
∴ The range of values of m is.
15. ∵ The equationhas real roots.
∴
i.e.
∴ The range of values of k is.
16. ∵ The equationhas real roots.
∴
i.e.
∴ The range of values of k is.
17. ∵ The graph of touches the x-axis at only one point.
∴
i.e.
18. ∵ The graph of cuts the x-axis at two distinct points.
∴
i.e.
∴ The range of values of m is.
Level 2
19.
∵ The equationhas one double real root.
∴
i.e.
20.
∵ The equationhas two distinct real roots.
∴
i.e.
∵ is a quadratic equation.
∴ The coefficient of x2 cannot be zero.
i.e.
∴ The range of values of m is except .
21. (a) ∵ The graph oftouches the x-axis at one point P.
∴
i.e.
(b) For k = 9, the corresponding quadratic equation is
∴ The coordinates of P are (–6, 0).
22. (a)
∵ The graph of touches the x-axis at one point Q.
∴
i.e.
(b) For, the corresponding quadratic equation is
∴ The coordinates of Q are.
∴ Length of OQ
23.
∵ The graph ofhas no x-intercepts.
∴
i.e.
∴ The range of values of p is.
24.
∵ The equationhas real roots.
∴
i.e.
∴ The largest value of k is.
25.
∵ The graph ofintersects the x-axis.
∴
i.e.
∴ The smallest value of m is.
26. (a)
∵ The graph of cuts the x-axis at two points.
∴
i.e.
∴ The range of values of p is.
(b) The largest integral value of p is 0.
For p = 0, the corresponding quadratic equation is
∴ The x-intercepts of the graph areand 1.
27. (a)
∵ The equationhas two unequal real roots.
∴
i.e.
∴ The range of values of k is.
(b) The smallest integral value of k is.
For, the corresponding quadratic equation is
28. For the equation,
∴ The equationhas two unequal real roots for any positive values of p.
29. ∵ The equationhas two distinct real roots.
∴
i.e.
∴ Any pair of integral values of a and c such that is acceptable.
∴
(or any other reasonable answers)
30. ∵ The graph oftouches the x-axis at one point.
∴
i.e.
∴ Any pair of integral values of m and n such that
m2 = 4n is acceptable.
∴
(or any other reasonable answers)
31. (a)
(i) ∵ The equationhas two distinct real roots.
∴
i.e.
∴ The range of possible values of k is
.
(ii) The only possible negative integral value of k is.
For, the corresponding quadratic equation is
(b)
Exercise 2B (p. 2.17)
Level 1
1. The required quadratic equation is
2. The required quadratic equation is
3. The required quadratic equation is
4. The required quadratic equation is
5. The required quadratic equation is
6.
∴ The required quadratic equation is
7.
∴ The required quadratic equation is
8.
∴ The required quadratic equation is
9.
∴ The required quadratic equation is
10.
∴ The required quadratic equation is
11.
∴ The required quadratic equation is
Level 2
12. (a)
(b) The roots of the required quadratic equation areand, i.e. 0 and 6 respectively.
∴ The required quadratic equation is
13. (a)
(b) The roots of the required quadratic equation are and, i.e. andrespectively.
∴ The required quadratic equation is
14. (a)
(b) The roots of the required quadratic equation are and, i.e.andrespectively.
∴ The required quadratic equation is
15. (a) Using the quadratic formula,
(b) The roots of the required quadratic equation are
, i.e. .
∴ The required quadratic equation is
16. (a)
Using the quadratic formula,
(b) The roots of the required quadratic equation are
, i.e. .
∴ The required quadratic equation is
17. (a) The required quadratic equation is
(b) When m = 1, n = 6 or m = 6, n = 1.
The required quadratic equation is
When m = 2, n = 3 or m = 3, n = 2.
The required quadratic equation is
Exercise 2C (p. 2.24)
Level 1
1.
2.
3.
4.
5. For the equation mx2 + 9x + n = 0,
6. Let a be the other root.
∴ The other root is 5.
7. (a)
(b)
8. (a)
(b)
9.
10.
When,
When,
∴ ,
11.
(a)
(b)
12.
(a)
(b)
13.
(a)
(b)
14. ∵ a and b are the roots of x2 – 5x – 3 = 0.
∴
(a) For the required quadratic equation,
∴ The required quadratic equation is
(b) For the required quadratic equation,
∴ The required quadratic equation is
15. ∵ a and b are the roots of .
∴
(a) For the required quadratic equation,
∴ The required quadratic equation is
(b) For the required quadratic equation,
∴ The required quadratic equation is
16. (a) Sum of roots =
Product of roots =
(b) ∵ –2 and 3 are the x-intercepts of the graph of
y = px2 + qx – 12.
∴ –2 and 3 are the roots of px2 + qx – 12 = 0.
17. (a) (i) By substituting (0, 15) into y = –x2 + mx + n,
we have
(ii) ∵ 5 is one of the x-intercepts of the graph of
y = –x2 + mx + n.
∴ 5 is one of the roots of –x2 + mx + n = 0.
Let a be the other root.
∴ The coordinates of P are (–3, 0).
(b)
Level 2
18.
19.
20. (a)
∵ Sum of roots = product of roots + 6
∴
(b) By substituting k = 5 into the equation, we have
21. Let a and be the roots of the equation.
22. Letandbe the roots of , where
.
∵ The difference between the roots is 3.
∴
23.
(a)
∴
(b)
24.
(a)
(b)
25.
(a)
(b)
26. Sum of roots =
Product of roots =
(a)
(b)
(c) From (b), (rejected
∵ )
(d)
27.
(a)
(b)
(c)
(d) From (c), (rejected
∵ )
28. ∵ a and b are the roots of .
∴
For the required quadratic equation,
∴ The required quadratic equation is
29. ∵ a and b are the roots of .
∴
For the required quadratic equation,
∴ The required quadratic equation is
30.