Solving Systems of Quadratic Equations
Understanding Systems of Quadratic Equations
This blog passage takes a gander at four frameworks of quadratic conditions.
In every framework, A, B, C, D, E, and F are constants and we are attempting to settle for x and y. Note: please confirm every arrangement except subbing the figured x and y back in the conditions.
Framework 1
A* x^2 + B* y = C
D* x^2 + E *y = F
A * x^2 + B * y = C
– A/D * (D * x^2 + E * y) = F * – A/D
I will unravel for y first.
A* x^2 + B * y = C
– A * x^2 – (A * E)/D * y = – (A * F)/D
(B – A * E/D) * y = C – (A * F)/D
y = (C – A * F/D)/(B – A * E/D)
Let y0 = y and now explain for x:
A * x^2 + B * y0 = C
A * x^2 = C – B * y0
x^2 = 1/A * (C – B * y0)
x = ±√( 1/A * (C – B * y0) )
Synopsis:
A* x^2 + B* y = C
D* x^2 + E *y = F
y = (C – A * F/D)/(B – A * E/D)
x = ±√( 1/A * (C – B * y) )
Model:
Pictures are produced by the HP Prime emulator.
x^2 + 5 * y = 10
4 * x^2 + 4/9 * y = 100
A = 1, B = 5, C = 10, D = 4, E = 4/9, F = 100
y = (C – A * F/D)/(B – A * E/D)
y = (10 – 1 * 100/4)/(5 – 1 * 4/9/4)
y = – 135/44 ≈ – 3.068181818
x^2 = ±√( 1/1 * (10 – 5 * – 135/44) )
x^2 = ±√( 1115/44 )
x ≈ ±5.033975476
The two focuses are:
( 5.033975476, – 3.0681818)
( – 5.033975476, – 3.0681818)
Framework 2
A * x^2 + B * y^2 = C
D * x^2 + E * y^2 = F
We’ll begin with fathoming for x:
A * x^2 + B * y^2 = C
– B * D/E * x^2 – B * y^2 = – B * F/E
(A – B * D/E) * x^2 = C – B * F/E
x^2 = (C – B * F/E)/(A – B * D/E)
x = ±√( (C – B * F/E)/(A – B * D/E) )
Let x0 = x
A * x0^2 + B * y^2 = C
B * y^2 = C – A * x0^2
y^2 = 1/B * (C – A * x0^2)
y = ±√( 1/B * (C – A * x0^2) )
Synopsis:
A * x^2 + B * y^2 = C
D * x^2 + E * y^2 = F
x = ±√( (C – B * F/E)/(A – B * D/E) )
y = ±√( 1/B * (C – A * x0^2) )
Model:
3 * x^2 + y^2 = 25/16
36 * x^2 + y^2 = 4
A = 3, B = 1, C = 25/16, D = 36, E = 1, F =4
x = ±√( (25/16 – 1 * 4/1)/(3 – 1 * 36/1) )
x = ±√( 13/176 ) ≈ ± 0.271778653
x^2 = 13/176
y = ±√( 1/1 * (25/16 – 3 * 13/176) )
y = ±√( 59/44 ) ≈ ± 1.157976291
Arrangements:
(0.271778653, 1.157976291)
(0.271778653, – 1.157976291)
(- 0.271778653, 1.157976291)
(- 0.271778653, – 1.157976291)
Framework 3
A * x^2 + B * y^2 = C
D * x^2 + E * y = F
A * x^2 + B * y^2 = C
– A/D * (D * x^2 + E * y) = F * – A/D
A * x^2 + B * y^2 = C
– A * x^2 + – A * E/D * y = – A * F/D
B * y^2 + – A * E/D * y = C – A * F/D
B * y^2 + – A * E/D * y + (A * F/D – C) = 0
When y is fathomed for, at that point:
A * x^2 = C – B * y^2
x^2 = 1/A * (C – B * y^2)
x = ±√ ( 1/A * (C – B * y^2) )
Rundown:
A * x^2 + B * y^2 = C
D * x^2 + E * y = F
B * y^2 + – A * E/D * y + (A * F/D – C) = 0
x = ±√ ( 1/A * (C – B * y^2) )
Model:
49/16 * x^2 + 16 * y^2 = 64
x^2 + 7 * y = 5
A = 49/16, B = 16, C = 4, D = 1, E = 7, F = 5
B = 16
– A * E/D = – 343/16
A * F/D – C = – 779/16
y1 ≈ – 1.19804377
x1 ≈ ±3.659362053
y2 ≈ 2.538548127
x2 ≈ ±3.573490854i
Arrangements (genuine numbers):
( 3.659362053, – 1.19804377)
( – 3.659362053, – 1.19804377)
Framework 4
A * x^2 + B * y^2 = C
D * x + E * y = F
E * y = F – D * x
y = F/E – D/E * x
y^2 = (D/E)^2 * x^2 – 2 * D * F/E^2 * x + (F/E)^2
A * x^2 + B * ((D/E)^2 * x^2 – 2 * D * F/E^2 * x + (F/E)^2) = C
A * x^2 + B * ((D/E)^2 * x^2 – 2 * D * F/E^2 * x + (F/E)^2) – C = 0
A * x^2 + B * (D/E)^2 * x^2 – (2 * B * D * F/E^2) * x + ( B * (F/E)^2 – C ) = 0
When the answers for x are found, illuminate for y by:
y = F/E – D/E * x
Outline:
A * x^2 + B * y^2 = C
D * x + E * y = F
(A + B * (D/E)^2) * x^2 + (- 2 * B * D * F/E^2) * x + ( B * (F/E)^2 – C ) = 0
y = F/E – D/E * x
Model 4:
5 * x^2 + 6 * y^2 = 15
– 10 * x + 13 * y = 2
A = 5, B = 6, C = 15, D = – 10, E = 13, F = 2
A + B * (D/E)^2 = 1445/169
– 2 * B * D * F/E^2 = 240/169
B * (F/E)^2 – C = – 2511/169
x1 ≈ 1.2537792907
y1 ≈ 1.105994544
x2 ≈ – 1.403882873
y2 ≈ – 0.926063748
I trust you locate this accommodating.
