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.

Leave a Reply

Your email address will not be published. Required fields are marked *