Question

Y=ax2+bx+c проходит через координаты A(-1,0) B(0,3) C(2,-3) найдите a b c

Answer 1

Подставим в уравнение y=ax2+bx+c вместо переменных х; у координаты данных точек A(-1;  0),  B(0;  3),  C(2;  -3) и получим систему трех уравнений с тремя переменными:

{0=a·(-1)²+b·(-1)+c

{3=a·0²+b·0+c

{-3=a·2²+b·2+c

Приведем к привычному виду:

{a-b+c=0

{c=3

{4a+2b+c=-3

Подставим с=3 в первое и третье уравнения и получим систему двух уравнений:

{a-b+3=0

{4a+2b+3=-3

Из первого уравнения выразим а через b

a=b-3

и подставим во второе:

4·(b-3)+2b+3 = -3

4b-12+2b+3= -3

6b = 6

b= 6:6

b=1         =>   a=b-3  => a= 1-3= -2

Ответ:   a= - 2;   b = 1;   c = 3

Answer 2

To find the values of the coefficients a, b, and c for the quadratic equation $y = ax^2 + bx + c$ that passes through the given points A(-1, 0), B(0, 3), and C(2, -3), we can create three equations using these points and solve for a, b, and c.

Let's start with point A(-1, 0): $0 = a(-1)^2 + b(-1) + c$

Next, point B(0, 3): $3 = a(0)^2 + b(0) + c$

Finally, point C(2, -3): $-3 = a(2)^2 + b(2) + c$

Now, let's solve these equations simultaneously to find the values of a, b, and c.

1. From the equation using point A: $0 = a - b + c \quad \quad \text{(Equation 1)}$

2. From the equation using point B: $3 = c \quad \quad \quad \quad \quad \quad \text{(Equation 2)}$

3. From the equation using point C: $-3 = 4a + 2b + c \quad \quad \text{(Equation 3)}$

Substitute the value of c from Equation 2 into Equations 1 and 3:

1. $0 = a - b + 3$
2. $-3 = 4a + 2b + 3$

Now, let's solve these two equations:

1. $a - b = -3 \quad \quad \quad \quad \text{(Equation 4)}$
2. $4a + 2b = -6 \quad \quad \quad \text{(Equation 5)}$

Multiply Equation 4 by 2 to make the coefficients of b in both equations equal:

$2a - 2b = -6 \quad \quad \quad \text{(Equation 6)}$

Add Equation 5 and Equation 6:

$2a - 2b + 4a + 2b = -6 - 6$

$6a = -12$

$a = -2$

Now that we have the value of a, we can find b from Equation 4:

$-2 - b = -3$

$b = -2 - (-3)$

$b = -2 + 3$

$b = 1$

Lastly, we already found that c = 3 from Equation 2.

So, the values of a, b, and c are:

$a = -2$

$b = 1$

$c = 3$