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SVETLANKA909090 [29]
3 years ago
14

The graph of $y=ax^2+bx+c$ passes through points $(0,5)$, $(1,10)$, and $(2,19)$. Find $a+b+c$.

Mathematics
1 answer:
MAVERICK [17]3 years ago
7 0

Answer:

a+b+c=10

Step-by-step explanation:

We are given that the graph of the equation:

y=ax^2+bx+c

Passes through the three points (0, 5), (1, 10), and (2, 19).

And we want to find the value of (<em>a</em> + <em>b</em> + <em>c</em>).

First, since the graph passes through (0, 5), its <em>y-</em>intercept or <em>c</em> is 5. Hence:

y=ax^2+bx+5

Next, since the graph passes through (1, 10), when <em>x</em> = 1, <em>y</em> = 10. Substitute:

(10)=a(1)^2+b(1)+5

Simplify:

5=a+b

The point (2, 19) tells us that when <em>x</em> = 2, <em>y</em> = 19. Substitute:

(19)=a(2)^2+b(2)+5

Simplify:

14=4a+2b

This yields a system of equations:

\begin{cases} 5 = a + b \\ 14 = 4a + 2b\end{cases}

Solve the system. We can do so using elimination (or any other method you prefer). Multiply the first equation by negative two:

-10=-2a-2b

Add the two equations together:

(-10)+(14)=(-2a+4a)+(-2b+2b)

Combine like terms:

4 = 2a

Hence:

a=2

Using the first equation:

5=(2)+b\Rightarrow b=3

Therefore, our equation is:

y=2x^2+3x+5

Thus, the value of (<em>a</em> + <em>b</em> + <em>c</em>) will be:

a+b+c = (2) + (3) + (5) = 10

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