EXAMPLES OF QUADRATIC EQUATIONS.

Example 1. Reduce 3x2

1. Transposing and uniting,

2. Divide by 3,

-9x-4-80.

3x2-9x=84.

x2-3x=28.

3. Comp. the square, (Art. 311. b.) 4x2-12x+9=112+9.

4. Extracting the square root,

5. Transposing and uniting,

Example 2. Reduce 4x

36- -X

2x-3=+11.

x=7, or -4.

-46

1. Clearing of fractions, 4x2-36+x=46x. 2. Transposing and uniting, 4x2 −45x=36.

3. Completing the square (Art. 311. b.) 64x2-720x+2025 =576+2025.

4. Extracting the square root, 8x-45 +51.

5. Transposing and uniting, x=12, or —3.

1. Clearing of fractions, 4x2+4x-14+x=14x+14.

2. Transposing and uniting, 4x2-9x=28.

3. Completing the square, (Art. 311, b.) 64x2-144x+81

=448+81.

4. Extracting the square root, 8x-9=±23.

5. Transposing and uniting, x=4, or −7.

Example 4. Reduce 5x-3x-3

3x-6

=2x+

X- -3

2

1. Clearing of fractions, 10x2-30x-6x+6=4x2-12x

+3x2-6x-9x+18.

2. Transposing, 10x2-4x2-3x2-30x-6x+12x+6x+9x =-6+18.

3. Uniting terms, and dividing each side by 3, x2—3x=4. 4. Completing the sq. (Art. 311, b.) 4x2-12x+9=9+16. 5. Extracting the square root, 2x−3=√25.

6. Transposing and uniting, x=4, or —1.

1. Clearing of fractions, 64x2 -100x+9x2=12x3. 2. Dividing by x, 64x-100+9x=12x2.

3. Transposing and uniting, -12x2+73x=100.

4. Changing signs, 12x2-73x=-100.

5. Completing the square, (Art. 311, b.) 576x2-3504x+ 5329-4800+5329

6. Extracting the square root, 24x-73=±23. 7. Transposing and uniting, x=4, or 2.

Example 6. Reduce

3x-4

=10—2—2. +1=10x

x-4

2

1. Cl. of fr. 6x÷8+2x-8=20x—80—x2+2x+4x−8. 2. Transposing and uniting, x2-18x=72.

3. Comp. square, (Art. 311. b.) x2-18x+81=-72+81. 4. Extracting the square root, x-9=±3, and x=

Example 7. Reduce +4_7-x

3 x-3

4x+7
9

-1.

=12, or 6.

1. C.F.9x2-27x+36x-108-189+27x=12x2 -36x+21x

-63-27x+81.

2. Transposing and uniting, -3x2+78x=315.

3. Changing signs and dividing by 3, x2-26x=-105. 4. Comp. square, (Art. 311. b.) 4x2-104x+676=-420 +676.

5. Extracting the square root, 2x-26=√256, and x=21,

or 5.

Example 8. Reduce.

x2-10x2+1

=x- -3.

x2-6x+9

1. Clearing of fractions, x3-10x2+1=x3—9x2 +27x—27. 2. Cancelling terms, 2+1=27x-27.

3. Transposing and changing signs, x2+27x=28.

4. Comp. square, (Art. 311. b.) 4x2 +108+729=112+729. 5. Extract the square root, 2x+27=±29.

6. Transposing and uniting, x=1, or -28.

1. Clearing of fractions, 6x+2x+2=3x2+3x. 2. Transposing and uniting, -3x2+5x=-2.

3. Changing signs, 3x2-5x-2.

4. Comp. the sq. (Art. 311. b.) 36x2 −60x+25=24+25. 5. Extracting the square root, 6x-5=√49=7, and x=2.

1. Clearing of fractions, 18x-x2+x-2x+2=6x2-54x+

12x-108.

2. Transposing and uniting, -7x2 +59x=-110. 3. Changing signs, 7x2 −59x=110.

4. Completing the square, (Art. 311. b.) 196x2-1652.x +3481 3080+3481.

5. Extracting the square root, 14x-59=±81, and x=10.

1. Clearing of fractions, ax2 +a3=2ax.

2. Dividing by a and transposing, x2 — 2x——a2.

3. Completing the square, (Art. 305.) x2-2x+1=1-a2. 4. Extracting the square root, x-1=√1-a2.

5. Transposing, x=11-a2.

Example 12. Reduce x4 +ax2=b.

1. Comp. square, (Art.314.) x4 +ax2+

a2

ስ?

=b+

4

4

a +

a2

b+

-2

32

23

1

2

16

2. Comp. square, (Art. 314.)x

3. Extracting the square root, x3—1=0. 4. Transposing, x3=4, andx=.

Example 14. Reduce 2x3+ 3x3—2.

1. Comp. square, (Art. 311. b.) 16x3+24x3 +9=16+9

2. Extracting the square root, 4x+3=5.

3. Transposing and uniting, 4x3=2. 4. Dividing by 4, and cubing, x=f.

Example 15. Reduce x-√x=22}.

1. Clearing of fractions and dividing, 3x-2√x=133. 2. Comp. square, (Art. 311. b.) 36x-24/x+4=1596+4. 3. Extracting the square root, 6x-2=40.

4. Tr. dividing by 6, and squaring, x=7, and x=49.

Example 16. Reduce 2x4x2+96=99.

1. Transposing and uniting, 2x4 —x2=3.

2. Comp. square, (Art. 311. b.) 16x4 −8x2+1=24+1.
3. Extracting the square root, 4x2-1=√25=5.
4. Transposing and evolving, 2x=√6, and x=1√6.

Example 17. Reduce (10+x)*— (10+x)‡—2.
1. Com. the sq. (Art. 311. b.&Art.308.) 4(10+2)—4
(10+x)+1=8+1

2. Extracting the square root, 2(10+x)¤—1—3 3. Transposing and uniting, 2(10+x) —4

4. Dividing by 2, (10+x)=2

5. Involving to the fourth power, 10+x=16, and x=6.

Example 18. Reduce 3x2"-2x”=8.

1. Completing the square, (Art. 311. b.) 36x2"-24x+4=

96+4

2. Extracting the square root, 6x-2=10, and 6x=12 3. Dividing by 6, and evolving, "=2, and x=”√/2.

Example 19. Reduce 2(1+x—x2)−√1+x−x2:

1. Divide by 2, (1+x−x2) — √1+x−x2

2

2. Comp. the square, (Art. 311. b.) (1+x-x2) — (1+x—x2),

3. Extracting the square root, ✓1+x¬x2−4=√1⁄2 4. Transposing and uniting,

2

5. Involving both sides and trans. x-x2=-1=-}

6. Changing signs,

7. Completing the square, (Art.305.) x2-x+=+&+= }} 3. Extracting the square root, x-√√41

9. Transposing,

Example 20. Reduce 3/x3-a3=x—b.

1. Involving each side to the 3d power, x3-a3x3—3bx2

+3b2x-b3

2. Transposing and uniting, -362x+3bx2-a3-b3

3. Dividing by 3b, -bx+x2

=

a3-b3
3b

=

4. Completing the square, (Art. 305.) x2 bx+

4a3-b3

126

1. Clearing of fractions, 2x+2x=16-x 2. Transposing and uniting, 2x=16-3x

3. Involving, 4x=256–96x+9x2

4. Transposing and uniting, -9x2 +100x=256

5. Changing signs, 9x2-100x=-256

6. Completing the square, (Art. 311. b.) 324x2-3600x+ 10000-9216+10000

7. Extracting the square root, 182—100——28 8. Transposing and uniting, 18x=72, and x=4.

6

3

Example 22. Reduce x3 + x5=756

1. Complete the square, (Art. 311. b.) 4x3+4x3+1= 3024+1

2. Extracting the square root, 2x3+1=55

3. Tr. unite and divide, 23–27

4. By evolution and involution, x=3, and x=243.

Example 23. Reduce 2x+1+2√x=·

21

√2x+1