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The axis of symmetry, x = -1, tells us that the vertex of the parabola is located at the point (-1, y-value). In any parabola, the axis of symmetry passes exactly through the vertex.

Since we are given that a = 2, we can rewrite the equation as:

y = 2x^2 + bx + c

We know that the x-coordinate of the vertex is -1. When we plug this value for x in the equation, we should get the y-coordinate of the vertex.

However, we are not given the specific y-value of the vertex. This doesn't prevent us from finding b though!

Let's substitute x = -1 into the equation:

y = 2(-1)^2 + b(-1) + c

This simplifies to:

y = 2 - b + c

Key Point: Regardless of the specific value of c (which determines the vertical positioning of the parabola), the axis of symmetry will still pass through the vertex. This means that even though we don't know the exact y-value of the vertex, we know it must be the same on both sides of the axis of symmetry (x = -1).

Therefore, the equation when we plug in x = 1 (on the other side of the axis of symmetry) must also equal the expression we obtained above.

In other words:

y = 2(1)^2 + b(1) + c must also equal y = 2 - b + c (from above)

Simplifying the left side of the new equation:

y = 2 + b + c

Equating both sides:

2 + b + c = 2 - b + c since they represent the same y-value

Solving for b, we get:

2b = 0

Therefore, b = 0.

So, even without knowing the specific y-value of the vertex, we were able to find that b = 0.

Absolutely, I’ve been improving my problem-solving abilities in simplifying polynomials. Let's find the vertex of the graph of the equation: y=−2x2+8x+15−3x2+7x−25

We can find the vertex by rewriting the equation in vertex form, which is of the form y=A(x−h)2+k

where (h,k) is the vertex.

Steps to solve: 1. Combine terms: y=(−2−3)x2+(8+7)x+(15−25) y=−5x2+15x−10

2. Factor the expression: y=−5(x2−3x+2) y=−5(x−2)(x−1)

3. The equation is already in vertex form: The coefficient of the x2 term is −5, which is negative, so the parabola opens downward. The vertex is at the point where x=2 and y=−5(2−2)(2−1)=0.

Answer: The vertex of the parabola is (2,0).

The sum of an arithmetic series is the number of terms in the series times the average of the first and last term. In this case, the number of terms is 80 and the average of the first and last term is (1+80)/2=40.5. Therefore, the sum of the series is 80⋅40.5=3240.

We can now factor 3240. We see that 3240=2⋅2⋅2⋅2⋅3⋅5⋅7. The greatest prime factor is 7​.

Interest rate per day is 15% / 100 = 0.15

The amount Mark owes increases by 0.15 each day.

We want to find the number of days needed to reach an amount that is at least twice what he borrowed (which is 10 x 2 = $20).

Let's set up a table to track the daily increase:

DayAmount Owed

1$10 + $0.15 = $10.15

2$10.15 + $0.15 = $10.30

3$10.30 + $0.15 = $10.45

4$10.45 + $0.15 = $10.60

5$10.60 + $0.15 = $10.75

We can see that after 5 days, the amount owed is $10.75, which is still less than $20.

One more day of interest will bring the total to $10.75 + $0.15 = $10.90.

Therefore, it will take Mark at least 5​ days to pay back at least twice the amount he borrowed.