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------------------------------------------------ Find an equation of the form [tex]f(x) = ax^2 + bx + c[/tex]. You must solve algebraically and check using a calculator. State the steps used to perform the check.

Given:
[tex]f(1) = 4[/tex]
[tex]f(2) = 13[/tex]
[tex]f(4) = 46[/tex]

Answer :

By using algebra resolution methods for systems of linear equations, we find that the equation of the form f(x) = a · x² + b · x + c that passes through the three points is equal to f(x) = (5 / 2) · x² + (3 / 2) · x.

How to determine the quadratic equation that passes through three points

In this problem we must determine the coefficients of a quadratic equation that passes through the points (x₁, y₁) = (1, 4), (x₂, y₂) = (2, 13) and (x₃, y₃) = (4, 46). First, we need to create a system of linear equations by substituting on y and x thrice:

(x₁, y₁) = (1, 4)

a · 1² + b · 1 + c = 4

a + b + c = 4

(x₂, y₂) = (2, 13)

a · 2² + b · 2 + c = 13

4 · a + 2 · b + c = 13

(x₃, y₃) = (4, 46)

a · 4² + b · 4 + c = 46

16 · a + 4 · b + c = 46

Then, we find a system of three linear equations with three variables that offers an unique solution:

a + b + c = 4 (1)

4 · a + 2 · b + c = 13 (2)

16 · a + 4 · b + c = 46 (3)

There are different methods to find the solution to this system, we proceed to use algebraic substitution:

By (1):

c = 4 - a - b

(1) in (2) and (3):

4 · a + 2 · b + (4 - a - b) = 13

3 · a + b = 9 (2b)

16 · a + 4 · b + (4 - a - b) = 46

15 · a + 3 · b = 42 (3b)

By (2b):

b = 9 - 3 · a

(2b) in (3b):

15 · a + 3 · (9 - 3 · a) = 42

15 · a + 27 - 9 · a = 42

6 · a = 15

a = 15 / 6

a = 5 / 2

By (2b):

b = 9 - 3 · (5 / 2)

b = 9 - 15 / 2

b = 18 / 2 - 15 / 2

b = 3 / 2

By (1):

c = 4 - 5 / 2 - 3 / 2

c = 4 - 8 / 2

c = 4 - 4

c = 0

The coefficients of the quadratic equation are (a, b, c) = (5 / 2, 3 / 2, 0).

To learn more on quadratic equations: https://brainly.com/question/1863222

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