Exercise algebra.equations.quadratic.approximate
Description
solve a quadratic equation with approximation
Derivations
1.
x^2 == 2
⇒ algebra.equations.coverup.powerx == sqrt 2 or x == -sqrt 2
⇒ algebra.equations.quadratic.approxx ~= 1.4142 or x ~= -1.4142
2.
x^2+3 == 52
⇒ algebra.equations.coverup.onevar.plusx^2 == 49
⇒ algebra.equations.coverup.powerx == 7 or x == -7
3.
x^2-7 == 0
⇒ algebra.equations.quadratic.no-linx^2 == 7
⇒ algebra.equations.coverup.powerx == sqrt 7 or x == -sqrt 7
⇒ algebra.equations.quadratic.approxx ~= 2.6458 or x ~= -2.6458
4.
9*x^2-6 == 75
⇒ algebra.equations.coverup.onevar.minus-left9*x^2 == 81
⇒ algebra.equations.coverup.timesx^2 == 9
⇒ algebra.equations.coverup.powerx == 3 or x == -3
5.
32-2*x^2 == 14
⇒ algebra.equations.coverup.onevar.minus-right2*x^2 == 18
⇒ algebra.equations.coverup.timesx^2 == 9
⇒ algebra.equations.coverup.powerx == 3 or x == -3
6.
2*(x^2-3) == 12
⇒ algebra.equations.coverup.timesx^2-3 == 6
⇒ algebra.equations.coverup.onevar.minus-leftx^2 == 9
⇒ algebra.equations.coverup.powerx == 3 or x == -3
7.
1/4*x^2+12 == 16
⇒ algebra.equations.coverup.onevar.plus1/4*x^2 == 4
⇒ algebra.equations.coverup.timesx^2 == 16
⇒ algebra.equations.coverup.powerx == 4 or x == -4
8.
(x-1)^2 == 100
⇒ algebra.equations.coverup.powerx-1 == 10 or x-1 == -10
⇒ algebra.equations.coverup.onevar.minus-leftx == 11 or x-1 == -10
⇒ algebra.equations.coverup.onevar.minus-leftx == 11 or x == -9
9.
14-2*x^2 == 6
⇒ algebra.equations.coverup.onevar.minus-right2*x^2 == 8
⇒ algebra.equations.coverup.timesx^2 == 4
⇒ algebra.equations.coverup.powerx == 2 or x == -2
10.
1/4*(17-x^2) == 2
⇒ algebra.equations.coverup.times17-x^2 == 8
⇒ algebra.equations.coverup.onevar.minus-rightx^2 == 9
⇒ algebra.equations.coverup.powerx == 3 or x == -3
11.
(x-7)^2+3 == 11
⇒ algebra.equations.coverup.onevar.plus(x-7)^2 == 8
⇒ algebra.equations.coverup.powerx-7 == sqrt 8 or x-7 == -sqrt 8
⇒ algebra.equations.coverup.onevar.minus-leftx == 7+sqrt 8 or x-7 == -sqrt 8
⇒ algebra.equations.coverup.onevar.minus-leftx == 7+sqrt 8 or x == 7-sqrt 8
⇒ algebra.equations.quadratic.approxx ~= 9.8284 or x ~= 4.1716
12.
(6-2*x)^2 == 81
⇒ algebra.equations.coverup.power6-2*x == 9 or 6-2*x == -9
⇒ algebra.equations.coverup.onevar.minus-right2*x == -3 or 6-2*x == -9
⇒ algebra.equations.coverup.timesx == -3/2 or 6-2*x == -9
⇒ algebra.equations.coverup.onevar.minus-rightx == -3/2 or 2*x == 15
⇒ algebra.equations.coverup.timesx == -3/2 or x == 15/2
13.
1/2*(x+9)^2 == 4
⇒ algebra.equations.coverup.times(x+9)^2 == 8
⇒ algebra.equations.coverup.powerx+9 == sqrt 8 or x+9 == -sqrt 8
⇒ algebra.equations.coverup.onevar.plusx == -9+sqrt 8 or x+9 == -sqrt 8
⇒ algebra.equations.coverup.onevar.plusx == -9+sqrt 8 or x == -9-sqrt 8
⇒ algebra.equations.quadratic.approxx ~= -6.1716 or x ~= -11.8284
14.
(3-x^2)/10 == 2
⇒ algebra.equations.coverup.numerator3-x^2 == 20
⇒ algebra.equations.coverup.onevar.minus-rightx^2 == -17
⇒ algebra.equations.coverup.powerfalse
15.
5*x^2+3*x == 3*x+2
⇒ algebra.equations.quadratic.cancel5*x^2 == 2
⇒ algebra.equations.coverup.timesx^2 == 2/5
⇒ algebra.equations.coverup.powerx == sqrt (2/5) or x == -sqrt (2/5)
⇒ algebra.equations.quadratic.approxx ~= 0.6325 or x ~= -0.6325
16.
11-(2*x+1)^2 == 5
⇒ algebra.equations.coverup.onevar.minus-right(2*x+1)^2 == 6
⇒ algebra.equations.coverup.power2*x+1 == sqrt 6 or 2*x+1 == -sqrt 6
⇒ algebra.equations.coverup.onevar.plus2*x == -1+sqrt 6 or 2*x+1 == -sqrt 6
⇒ algebra.equations.coverup.timesx == (-1+sqrt 6)/2 or 2*x+1 == -sqrt 6
⇒ algebra.equations.coverup.onevar.plusx == (-1+sqrt 6)/2 or 2*x == -1-sqrt 6
⇒ algebra.equations.coverup.timesx == (-1+sqrt 6)/2 or x == (-1-sqrt 6)/2
⇒ algebra.equations.quadratic.approxx ~= 0.7247 or x ~= -1.7247
17.
(6*x-3)^2+6 == 12
⇒ algebra.equations.coverup.onevar.plus(6*x-3)^2 == 6
⇒ algebra.equations.coverup.power6*x-3 == sqrt 6 or 6*x-3 == -sqrt 6
⇒ algebra.equations.coverup.onevar.minus-left6*x == 3+sqrt 6 or 6*x-3 == -sqrt 6
⇒ algebra.equations.coverup.timesx == (3+sqrt 6)/6 or 6*x-3 == -sqrt 6
⇒ algebra.equations.coverup.onevar.minus-leftx == (3+sqrt 6)/6 or 6*x == 3-sqrt 6
⇒ algebra.equations.coverup.timesx == (3+sqrt 6)/6 or x == (3-sqrt 6)/6
⇒ algebra.equations.quadratic.approxx ~= 0.9082 or x ~= 9.18e-2
18.
(7+2*x)^2 == 7
⇒ algebra.equations.coverup.power7+2*x == sqrt 7 or 7+2*x == -sqrt 7
⇒ algebra.equations.coverup.onevar.plus2*x == -7+sqrt 7 or 7+2*x == -sqrt 7
⇒ algebra.equations.coverup.timesx == (-7+sqrt 7)/2 or 7+2*x == -sqrt 7
⇒ algebra.equations.coverup.onevar.plusx == (-7+sqrt 7)/2 or 2*x == -7-sqrt 7
⇒ algebra.equations.coverup.timesx == (-7+sqrt 7)/2 or x == (-7-sqrt 7)/2
⇒ algebra.equations.quadratic.approxx ~= -2.1771 or x ~= -4.8229
19.
4-x^2/10 == 6
⇒ algebra.equations.coverup.onevar.minus-rightx^2/10 == -2
⇒ algebra.equations.coverup.numeratorx^2 == -20
⇒ algebra.equations.coverup.powerfalse
20.
12-(2*x+3)^2 == 6
⇒ algebra.equations.coverup.onevar.minus-right(2*x+3)^2 == 6
⇒ algebra.equations.coverup.power2*x+3 == sqrt 6 or 2*x+3 == -sqrt 6
⇒ algebra.equations.coverup.onevar.plus2*x == -3+sqrt 6 or 2*x+3 == -sqrt 6
⇒ algebra.equations.coverup.timesx == (-3+sqrt 6)/2 or 2*x+3 == -sqrt 6
⇒ algebra.equations.coverup.onevar.plusx == (-3+sqrt 6)/2 or 2*x == -3-sqrt 6
⇒ algebra.equations.coverup.timesx == (-3+sqrt 6)/2 or x == (-3-sqrt 6)/2
⇒ algebra.equations.quadratic.approxx ~= -0.2753 or x ~= -2.7247
21.
x^2 == 5*x
⇒ algebra.equations.quadratic.move-leftx^2-5*x == 0
⇒ algebra.equations.quadratic.common-factorx*(x-5) == 0
⇒ algebra.equations.quadratic.product-zerox == 0 or x == 5
22.
x^2-6*x == 0
⇒ algebra.equations.quadratic.common-factorx*(x-6) == 0
⇒ algebra.equations.quadratic.product-zerox == 0 or x == 6
23.
6*x+x^2 == 0
⇒ algebra.equations.quadratic.common-factorx*(x+6) == 0
⇒ algebra.equations.quadratic.product-zerox == 0 or x == -6
24.
25.
x*(2*x-4) == 0
⇒ algebra.equations.quadratic.product-zerox == 0 or 2*x-4 == 0
⇒ algebra.equations.coverup.onevar.minus-leftx == 0 or 2*x == 4
⇒ algebra.equations.coverup.timesx == 0 or x == 2
26.
3*x^2 == 6*x
⇒ algebra.equations.quadratic.same-con-factorx^2 == 2*x
⇒ algebra.equations.quadratic.move-leftx^2-2*x == 0
⇒ algebra.equations.quadratic.common-factorx*(x-2) == 0
⇒ algebra.equations.quadratic.product-zerox == 0 or x == 2
27.
3*x == 2*x^2
⇒ algebra.equations.quadratic.move-left-2*x^2+3*x == 0
⇒ algebra.equations.quadratic.common-factor-x*(2*x-3) == 0
⇒ algebra.equations.quadratic.product-zerox == 0 or 2*x-3 == 0
⇒ algebra.equations.coverup.onevar.minus-leftx == 0 or 2*x == 3
⇒ algebra.equations.coverup.timesx == 0 or x == 3/2
28.
x*(1-6*x) == 0
⇒ algebra.equations.quadratic.product-zerox == 0 or 1-6*x == 0
⇒ algebra.equations.coverup.onevar.minus-rightx == 0 or 6*x == 1
⇒ algebra.equations.coverup.timesx == 0 or x == 1/6
29.
30.
(3*x-1)*(x+3) == 0
⇒ algebra.equations.quadratic.product-zero3*x-1 == 0 or x == -3
⇒ algebra.equations.coverup.onevar.minus-left3*x == 1 or x == -3
⇒ algebra.equations.coverup.timesx == 1/3 or x == -3
31.
x^2-2*x == 3
⇒ algebra.equations.quadratic.move-leftx^2-2*x-3 == 0
⇒ algebra.equations.quadratic.nice-factors(x+1)*(x-3) == 0
⇒ algebra.equations.quadratic.product-zerox == -1 or x == 3
32.
x^2+12*x+20 == 0
⇒ algebra.equations.quadratic.nice-factors(x+2)*(x+10) == 0
⇒ algebra.equations.quadratic.product-zerox == -2 or x == -10
33.
x^2-x == 30
⇒ algebra.equations.quadratic.move-leftx^2-x-30 == 0
⇒ algebra.equations.quadratic.nice-factors(x+5)*(x-6) == 0
⇒ algebra.equations.quadratic.product-zerox == -5 or x == 6
34.
x*(x+2) == 8
⇒ algebra.equations.linear.distr-timesx^2+2*x == 8
⇒ algebra.equations.quadratic.move-leftx^2+2*x-8 == 0
⇒ algebra.equations.quadratic.nice-factors(x-2)*(x+4) == 0
⇒ algebra.equations.quadratic.product-zerox == 2 or x == -4
35.
x*(x-3) == 4
⇒ algebra.equations.linear.distr-timesx^2-3*x == 4
⇒ algebra.equations.quadratic.move-leftx^2-3*x-4 == 0
⇒ algebra.equations.quadratic.nice-factors(x+1)*(x-4) == 0
⇒ algebra.equations.quadratic.product-zerox == -1 or x == 4
36.
2*x+15 == x^2
⇒ algebra.equations.quadratic.move-left-x^2+2*x+15 == 0
⇒ algebra.equations.quadratic.simpler-polyx^2-2*x-15 == 0
⇒ algebra.equations.quadratic.nice-factors(x+3)*(x-5) == 0
⇒ algebra.equations.quadratic.product-zerox == -3 or x == 5
37.
4*x == 12-x^2
⇒ algebra.equations.quadratic.move-leftx^2+4*x-12 == 0
⇒ algebra.equations.quadratic.nice-factors(x-2)*(x+6) == 0
⇒ algebra.equations.quadratic.product-zerox == 2 or x == -6
38.
x^2 == 15-8*x
⇒ algebra.equations.quadratic.move-leftx^2+8*x-15 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == 124, a == 1, b == 8, c == -15]x == (-8+sqrt 124)/2 or x == (-8-sqrt 124)/2
⇒ algebra.equations.quadratic.approxx ~= 1.5678 or x ~= -9.5678
39.
x^2-9*x+18 == 0
⇒ algebra.equations.quadratic.nice-factors(x-3)*(x-6) == 0
⇒ algebra.equations.quadratic.product-zerox == 3 or x == 6
40.
x^2+14*x+24 == 0
⇒ algebra.equations.quadratic.nice-factors(x+2)*(x+12) == 0
⇒ algebra.equations.quadratic.product-zerox == -2 or x == -12
41.
(3*x+5)^2+(x-5)^2 == 40
⇒ algebra.equations.quadratic.distr-square9*x^2+30*x+25+(x-5)^2 == 40
⇒ algebra.equations.quadratic.distr-square9*x^2+30*x+25+x^2-10*x+25 == 40
⇒ algebra.equations.linear.merge10*x^2+20*x+50 == 40
⇒ algebra.equations.quadratic.same-con-factorx^2+2*x+5 == 4
⇒ algebra.equations.quadratic.move-leftx^2+2*x+1 == 0
⇒ algebra.equations.quadratic.nice-factors(x+1)^2 == 0
⇒ algebra.equations.coverup.powerx+1 == 0
⇒ algebra.equations.coverup.onevar.plusx == -1
42.
4*(10-x^2) == -2*x*(2*x+10)
⇒ algebra.equations.quadratic.same-con-factor2*(10-x^2) == -x*(2*x+10)
⇒ algebra.equations.linear.distr-times20-2*x^2 == -2*x^2-10*x
⇒ algebra.equations.quadratic.cancel20 == -10*x
⇒ algebra.equations.coverup.times-2 == x
⇒ algebra.equations.linear.flipx == -2
43.
x*(x+12) == 2*x^2
⇒ algebra.equations.linear.distr-timesx^2+12*x == 2*x^2
⇒ algebra.equations.quadratic.move-left-x^2+12*x == 0
⇒ algebra.equations.quadratic.common-factor-x*(x-12) == 0
⇒ algebra.equations.quadratic.product-zerox == 0 or x == 12
44.
3*(x-2)*(x+6) == 12*x
⇒ algebra.equations.quadratic.same-con-factor(x-2)*(x+6) == 4*x
⇒ algebra.equations.linear.distr-timesx^2+4*x-12 == 4*x
⇒ algebra.equations.quadratic.cancelx^2-12 == 0
⇒ algebra.equations.quadratic.no-linx^2 == 12
⇒ algebra.equations.coverup.powerx == sqrt 12 or x == -sqrt 12
⇒ algebra.equations.quadratic.approxx ~= 3.4641 or x ~= -3.4641
45.
8*x^2+4*x-24 == (x+3)*(x-8)
⇒ algebra.equations.linear.distr-times8*x^2+4*x-24 == x^2-5*x-24
⇒ algebra.equations.quadratic.cancel8*x^2+4*x == x^2-5*x
⇒ algebra.equations.quadratic.move-left7*x^2+9*x == 0
⇒ algebra.equations.quadratic.common-factorx*(7*x+9) == 0
⇒ algebra.equations.quadratic.product-zerox == 0 or 7*x+9 == 0
⇒ algebra.equations.coverup.onevar.plusx == 0 or 7*x == -9
⇒ algebra.equations.coverup.timesx == 0 or x == -9/7
46.
3*x^2-11 == (3+2*x)^2
⇒ algebra.equations.quadratic.distr-square3*x^2-11 == 9+12*x+4*x^2
⇒ algebra.equations.quadratic.move-left-x^2-12*x-20 == 0
⇒ algebra.equations.quadratic.simpler-polyx^2+12*x+20 == 0
⇒ algebra.equations.quadratic.nice-factors(x+2)*(x+10) == 0
⇒ algebra.equations.quadratic.product-zerox == -2 or x == -10
47.
2*x*(x-3)-3 == (x+2)*(x+6)
⇒ algebra.equations.linear.distr-times2*x^2-6*x-3 == x^2+8*x+12
⇒ algebra.equations.quadratic.move-leftx^2-14*x-15 == 0
⇒ algebra.equations.quadratic.nice-factors(x+1)*(x-15) == 0
⇒ algebra.equations.quadratic.product-zerox == -1 or x == 15
48.
12*(x^2-3*x)+8 == 56
⇒ algebra.equations.coverup.onevar.plus12*(x^2-3*x) == 48
⇒ algebra.equations.coverup.timesx^2-3*x == 4
⇒ algebra.equations.quadratic.move-leftx^2-3*x-4 == 0
⇒ algebra.equations.quadratic.nice-factors(x+1)*(x-4) == 0
⇒ algebra.equations.quadratic.product-zerox == -1 or x == 4
49.
4*x^2-6*x == x^2+9
⇒ algebra.equations.quadratic.move-left3*x^2-6*x-9 == 0
⇒ algebra.equations.quadratic.simpler-polyx^2-2*x-3 == 0
⇒ algebra.equations.quadratic.nice-factors(x+1)*(x-3) == 0
⇒ algebra.equations.quadratic.product-zerox == -1 or x == 3
50.
(x+1)*(x-5) == (x-2)*(x-3)
⇒ algebra.equations.linear.distr-timesx^2-4*x-5 == x^2-5*x+6
⇒ algebra.equations.quadratic.cancel-4*x-5 == -5*x+6
⇒ algebra.equations.quadratic.move-leftx-11 == 0
⇒ algebra.equations.coverup.onevar.minus-leftx == 11
51.
x^2+4*x-4 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == 32, a == 1, b == 4, c == -4]x == (-4+sqrt 32)/2 or x == (-4-sqrt 32)/2
⇒ algebra.equations.quadratic.approxx ~= 0.8284 or x ~= -4.8284
52.
x^2-6*x == 4
⇒ algebra.equations.quadratic.move-leftx^2-6*x-4 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == 52, a == 1, b == -6, c == -4]x == (6+sqrt 52)/2 or x == (6-sqrt 52)/2
⇒ algebra.equations.quadratic.approxx ~= 6.6056 or x ~= -0.6056
53.
x^2-12*x+34 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == 8, a == 1, b == -12, c == 34]x == (12+sqrt 8)/2 or x == (12-sqrt 8)/2
⇒ algebra.equations.quadratic.approxx ~= 7.4142 or x ~= 4.5858
54.
2*x^2+4*x-8 == 0
⇒ algebra.equations.quadratic.simpler-polyx^2+2*x-4 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == 20, a == 1, b == 2, c == -4]x == (-2+sqrt 20)/2 or x == (-2-sqrt 20)/2
⇒ algebra.equations.quadratic.approxx ~= 1.2361 or x ~= -3.2361
55.
(x-4)*(x-1) == 11
⇒ algebra.equations.linear.distr-timesx^2-5*x+4 == 11
⇒ algebra.equations.quadratic.move-leftx^2-5*x-7 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == 53, a == 1, b == -5, c == -7]x == (5+sqrt 53)/2 or x == (5-sqrt 53)/2
⇒ algebra.equations.quadratic.approxx ~= 6.1401 or x ~= -1.1401
56.
(x-7/2)^2 == 2*(x+4)
⇒ algebra.equations.quadratic.distr-squarex^2-7*x+49/4 == 2*(x+4)
⇒ algebra.equations.linear.distr-timesx^2-7*x+49/4 == 2*x+8
⇒ algebra.equations.quadratic.move-leftx^2-9*x+17/4 == 0
⇒ algebra.equations.quadratic.simpler-poly4*x^2-36*x+17 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == 1024, a == 4, b == -36, c == 17]x == 17/2 or x == 1/2
57.
x^2-3*x == 3*(x-2)
⇒ algebra.equations.linear.distr-timesx^2-3*x == 3*x-6
⇒ algebra.equations.quadratic.move-leftx^2-6*x+6 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == 12, a == 1, b == -6, c == 6]x == (6+sqrt 12)/2 or x == (6-sqrt 12)/2
⇒ algebra.equations.quadratic.approxx ~= 4.7321 or x ~= 1.2679
58.
(4-x)*(1-x) == 3*x
⇒ algebra.equations.linear.distr-times4-5*x+x^2 == 3*x
⇒ algebra.equations.quadratic.move-leftx^2-8*x+4 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == 48, a == 1, b == -8, c == 4]x == (8+sqrt 48)/2 or x == (8-sqrt 48)/2
⇒ algebra.equations.quadratic.approxx ~= 7.4641 or x ~= 0.5359
59.
2*x^2 == x*(x+2)+7
⇒ algebra.equations.linear.distr-times2*x^2 == x^2+2*x+7
⇒ algebra.equations.quadratic.move-leftx^2-2*x-7 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == 32, a == 1, b == -2, c == -7]x == (2+sqrt 32)/2 or x == (2-sqrt 32)/2
⇒ algebra.equations.quadratic.approxx ~= 3.8284 or x ~= -1.8284
60.
(1-x)^2 == x+2
⇒ algebra.equations.quadratic.distr-square1-2*x+x^2 == x+2
⇒ algebra.equations.quadratic.move-leftx^2-3*x-1 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == 13, a == 1, b == -3, c == -1]x == (3+sqrt 13)/2 or x == (3-sqrt 13)/2
⇒ algebra.equations.quadratic.approxx ~= 3.3028 or x ~= -0.3028