SECTION 3.1
ALGEBRAIC AND GRAPHICAL SOLUTIONS OF EQUATIONS 135
Figure 6 shows the graph of the equation y = x3 — 6x2 + 9x — \jx in the viewing
rectangle [1, 6] by [—2,10]. There are two x-intercepts in this viewing rectangle;
zooming in we see that the solutions are x = 2.18 and x ~ 3.72.
FIGURE 6
The equation in Example 4 actually has four solutions. You are asked to find the
other two in Exercise 27.
3.1
EXERCISES
1-14 • Solve the equation both algebraically and graphically.
1. 2*-9 = 5 \. 3x + 1 = 31
3. x -T^ 5x + 12
4. 6x + 15 = -3x
5. jx - 3 = 6 + 2*
= -f x + 14
6. |jc + 2(x
7. - + -- = 7
x
2x
8.
x+2
2x
9. x 2 - 32 = 0
11. 16x4 = 625
12. 2x5 - 243 = 0
13. (x - 5)4 - 80 = 0
14. 6(x + 2)5 = 64
j 15-22 • Solve the equation graphically in the given interval.
State each answer correct to two decimals.
[0,6]
16. x2 - Q.15x + 0.125 = 0; [-2, 2]
17. x3-6x2 + I L c - 6 = 0; [-1,4]
18. 16x 3 + I6x2 = x+ 1; [-2,2]
20. 1 + -Jx = >/l + x2;
22. jt'/ 2 + jt 1 / 3 -Jt = 0;
[-1,5]
23-26 • Find all real solutions of the equation, correct to two
decimals.
23. *3 - 2x2 - x - 1 = 0
24. x4 - 8x2 + 2 = 0
25. x(x - 1)(* + 2) = £jt
26. x4 = 16 - x3
10. x3 + 16 = 0
_,
19. x - Jx + 1 = 0;
[-3,3]
g 27. In Example 4 we found two solutions of the equation
x3 — 6x2 + 9x = \jx, the solutions that lie between 1
and 6. Find two more solutions, correct to two decimals.
2*
15. * 2 -7;c+ 12 = 0;
21. x 1 / 3 -;t = 0;
[-1,5]
[-1,5]
DISCOVERY • DISCUSSION
28. Algebraic and Graphical Solution Methods Write a
short essay comparing the algebraic and graphical methods
for solving equations. Make up your own examples to illustrate the advantages and disadvantages of each method.
j 29. How Many Solutions? This exercise deals with the
family of equations x3 — 3x = k.
(a) Draw the graphs of yi = x3 — 3x and y2 = k in the
same viewing rectangle, in the cases k = — 4, — 2,
0, 2, and 4. How many solutions of the equation
x3 — 3x = k are there in each case? Find the solutions
correct to two decimals.
(b) For what ranges of values of k does the equation have
one solution? two solutions? three solutions?