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vlada-n [284]
2 years ago
15

Find the smallest possible value of a quadratic equation

Mathematics
1 answer:
Lelechka [254]2 years ago
5 0

I'm assuming that you mean to find the minimum of a parabola, i.e. the minimum of a function defined as

f(x) = ax^2+bx+c,\quad a,b,c \in\mathbb{R},\quad a\neq 0

To find the minimum of a function, we have to find a point x_0 such that

f'(x_0) = 0,\quad f''(x_0) > 0

The first derivative is

f'(x) = 2ax+b \implies f'(x)=0 \iff x = \dfrac{-b}{2a}

The second derivative is

f''(x) = 2a

So, a parabola has a minimum only if a>0 (otherwise, the parabola is concave down and it has no lower bound). In that case, the minimum has coordinates

x = \dfrac{-b}{2a},\quad y = f\left(\dfrac{-b}{2a}\right) = a\left(\dfrac{-b}{2a}\right)^2 + b\left(\dfrac{-b}{2a}\right) + c = \dfrac{4ac-b^2}{4a}

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Evaluate this integral. I'm not sure how to evaluate integrals with absolute values.
SOVA2 [1]

Answer:

  1

Step-by-step explanation:

The first step is to understand what the integrand looks like. (See the graph below.)

It is -1 for x < 0 and +1 for x > 0. Thus, the value of this integral is ...

  \int\limits^0_{-1} {-1} \, dx + \int\limits^2_0 {1} \, dx=-1+2=1

_____

Essentially, you treat the absolute value as a piecewise linear function. This is the same way you treat an absolute value in any equation or inequality.

Here, this means you divide the integral into two parts: one where the integrand is -x/x, and one where it is +x/x.

6 0
3 years ago
25 points to anybody who can solve this pls!
Svetach [21]

Answer:

Q' (2,4)

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S' (5,3)

Step-by-step explanation:

When reflected across x-axis, the x-values stay the same but y-values change signs. Because you count how many points till the reflection line then same amount you have to go to the other side. But since it is saying x-axis, the reflection line is on x=0 so you just change the sign.

6 0
2 years ago
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Lena [83]

Answer:

62 1/2

Step-by-step explanation:

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The first step in the process for factoring the trinomial x^2-3x-40 is to
jenyasd209 [6]

(x-8)(x+5)

First thing you do is put the X's into the parentheticals because it is x^2 meaning there are 2 factors with x's. Then, you wonder what factors of -40 could add to get -3. In this case, they are -8 and 5. -8 times 5 is -40 and -8 +5 is -3. Then, you just add -8 and 5 beside the x's in the parentheticals.

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Find the slope of the given points (6, -10) and (-15, 15). Show your work.
charle [14.2K]

Answer:

Step-by-step explanation:

The slope of a line between two points can be calculated by dividing the difference of y values by the difference of x values

in this case

m = \frac{y_2-y_1}{x_2-x_1} = \frac{15-(-10)}{-15-6} = \frac{25}{-21} = \frac{-25}{21}

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3 years ago
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