1914 0I1 au of oldizzog uelle soilgub the quadratic formula, 4a3 1 -b ±,b2 + 2 y3 27 RESTAN $7 from which y and then x can be determined. Use this method to find a root of the cubics x3 + 81x = 702 and +6x2 +18x +13 = 0. [Hint: /142,884 = 378.] IS 13. By making the substitution x = y +5/y, find a root of the cubic equation x3= 15x 126. 14. Use Cardan's formula to find, in these examples of the irreducible case in cubics, a root of the given equations. (a) x= 63x +162 [Hint: 81 30/-3 = (-3 ±2/-3)3.] CU (b) x3= 7x +6. 3 10 3 -3 L21 (c) x6- 2x2 +5x. Hint: 3 + -3 2 11 nolo -3 = 3 28 5 3 5 --3 Hint: + 27 6 15. The great Persian poet, Omar Khayyam (circa 1050-1130), found a geometric solution of the cubic equation x3a2x = b by using a pair of intersecting conic sections. In modern notation, he first constructed the parabola x2 diameter AC = b/a2 on the x-axis, and let P be the point of intersection of the semicircle with the parabola (see the figure). A perpendicular is dropped = ay. Then he drew a semicircle with from P to the r-axis to nroduce a nint QUNCKN

Question

Number 13

1914
0I1
au of oldizzog uelle
soilgub
the quadratic formula,
4a3
1
-b ±,b2 +
2
y3
27
RESTAN
$7
from which y and then x can be determined. Use this
method to find a root of the cubics x3 + 81x = 702 and
+6x2 +18x +13 = 0. [Hint: /142,884 = 378.]
IS
13. By making the substitution x = y +5/y, find a root of
the cubic equation x3= 15x 126.
14. Use Cardan's formula to find, in these examples of the
irreducible case in cubics, a root of the given equations.
(a) x= 63x +162
[Hint: 81 30/-3 = (-3 ±2/-3)3.]
CU
(b) x3= 7x +6.
3
10
3
-3
L21
(c) x6- 2x2 +5x.
Hint: 3 +
-3
2
11
nolo
-3 =
3
28 5
3
5
--3
Hint:
+
27
6
15. The great Persian poet, Omar Khayyam (circa
1050-1130), found a geometric solution of the cubic
equation x3a2x = b by using a pair of intersecting
conic sections. In modern notation, he first constructed
the parabola x2
diameter AC = b/a2 on the x-axis, and let P be the
point of intersection of the semicircle with the
parabola (see the figure). A perpendicular is dropped
= ay. Then he drew a semicircle with
from P to the r-axis to nroduce a nint
QUNCKN
Expand
Transcribed Image Text

1914 0I1 au of oldizzog uelle soilgub the quadratic formula, 4a3 1 -b ±,b2 + 2 y3 27 RESTAN $7 from which y and then x can be determined. Use this method to find a root of the cubics x3 + 81x = 702 and +6x2 +18x +13 = 0. [Hint: /142,884 = 378.] IS 13. By making the substitution x = y +5/y, find a root of the cubic equation x3= 15x 126. 14. Use Cardan's formula to find, in these examples of the irreducible case in cubics, a root of the given equations. (a) x= 63x +162 [Hint: 81 30/-3 = (-3 ±2/-3)3.] CU (b) x3= 7x +6. 3 10 3 -3 L21 (c) x6- 2x2 +5x. Hint: 3 + -3 2 11 nolo -3 = 3 28 5 3 5 --3 Hint: + 27 6 15. The great Persian poet, Omar Khayyam (circa 1050-1130), found a geometric solution of the cubic equation x3a2x = b by using a pair of intersecting conic sections. In modern notation, he first constructed the parabola x2 diameter AC = b/a2 on the x-axis, and let P be the point of intersection of the semicircle with the parabola (see the figure). A perpendicular is dropped = ay. Then he drew a semicircle with from P to the r-axis to nroduce a nint QUNCKN

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