The parabola with equation $y=ax^2+bx+c$ is graphed below: (attached)

The zeros of the quadratic $ax^2 + bx + c$ are at $x=m$ and $x=n$, where $m>n$. What is $m-n$?

Step-by-step explanation:

Instead of using the standard form, we can use the vertex form of a quadratic equation:

Where a is the leading coefficient, and (h, k) is our vertex.

Our vertex point is at (2, -4). So, let’s substitute 2 for h and -4 for k:

Now, we need to determine a.

We know that it passes through the point (4, 12). So, when x is 4, y must be 12. In other words:

Solve for a. Subtract within the parentheses:

Add 4 to both sides:

Square:

Solve:

Thererfore, the value of a is 4.

So, our function is:

Now, let’s find our roots. Set the equation to 0 and solve for x:

So, our roots are 1 and 3.

The greater root is 3 and the lesser root is 1.

Therefore, m-n, where m>n, is 3-1 or 2.

Our final answer is 2.

2

Step-by-step explanation:

Source: AOPS

A)  both functions are linear

B) f(x) = 7446.4x + 178713.6 and f(x) = 11000x + 179000

C) House 2 will value $106894.4 more than house 1.

Step-by-step explanation:

A) Value Increase from year 1 to year 2:

House 1: 193,606.40 - 186,160 = 7446.4

House 2: 201,000 - 190,000 = 11000

Value Increase from year 2 to year 3:

House 1: 201,350.66 - 193,606.40 = 7744.26

House 2: 212,000 - 201,000 = 11000  

This means that a constant increament in x variable gives a constant increament in both houses vales. Then, both functions are linear.

B) The slope is the same as the value increment from one year to the next one.

slope (m) of House 1: 7446.4

slope (m) of House 2: 11000

General formula of a line:

f(x) = mx+b

Replacing with a known point:

House 1

186,160 = 7446.4(1) + b

b = 186,160 - 7446.4 = 178713.6

equation: f(x) = 7446.4x + 178713.6

House 2

190,000 = 11000(1) + b

b = 190,000 - 11000 = 179000

equation: f(x) = 11000x + 179000

C) Replacing x = 30 into each equaiton:

Value of House 1 after 30 years

f(x) = 7446.4(30) + 178713.6 = 402105.6

Value of House 2 after 30 years

f(30) = 11000(30) + 179000 = 509000

Then, house 2 will value 509000 - 402105.6 = $106894.4  more than house 1.