CALCU-4. 0.25 的算术平方根是

4. 0.25 的算术平方根是

Solution by Steps

step 1

To find the arithmetic square root of 0.25, we look for a number that, when multiplied by itself, gives 0.25

step 2

The asksia-ll calculator input is: square+root+0.25

step 3

The asksia-ll calculator computes the square root and provides the result:

\sqrt{0.25} = 0.5

step 4

The result can also be expressed as a fraction:

0.5 = \frac{1}{2}

step 5

The result in words is 'one half'

step 6

The asksia-ll calculator also provides the result in exponential form:

0.5 e^0 ≈ 0.50000

(real, principal root)

Answer

0.5 or

\frac{1}{2}

Key Concept

Finding the square root of a number

Explanation

The square root of a number is the value that, when multiplied by itself, gives the original number. For 0.25, the square root is 0.5, which can also be expressed as

\frac{1}{2}

or 'one half'.

24. 在平面直角坐标系

\mathrm{xOy}

中, 如果抛物线

y=a x^{2}+b x+c

上存在一点

\mathrm{A}

, 使点

\mathrm{A}

关于坐标原点

\mathrm{O}

的对称点

A^{\prime}

也在这条抛物线上, 那么我们把这条抛物线叫做回归抛物线,点

\mathrm{A}

叫做这条抛物线的回归点.

Solution by Steps

step 1

To find the regression point A on the parabola

y = ax^2 + bx + c

, we need to find a point

(x, y)

such that its reflection

A'

across the origin

O

is also on the parabola

step 2

The coordinates of

A'

are

(-x, -y)

since it is the reflection of

A

across the origin

step 3

For

A'

to be on the parabola, the equation

y = ax^2 + bx + c

must also be satisfied by

(-x, -y)

step 4

Substituting

(-x, -y)

into the parabola equation gives us

-y = a(-x)^2 + b(-x) + c

step 5

Simplifying the equation from step 4, we get

-y = ax^2 - bx + c

step 6

Since

A

is on the parabola, we have

y = ax^2 + bx + c

. To find the regression point, we set

y

from the equation of

A

equal to

-y

from the equation of

A'

step 7

Setting

y = -y

gives us

ax^2 + bx + c = -(ax^2 - bx + c)

step 8

Simplifying the equation from step 7, we get

2ax^2 + 2bx = 0

step 9

Factoring out

2x

gives us

2x(ax + b) = 0

step 10

Setting each factor equal to zero gives us the solutions

x = 0

or

ax + b = 0

step 11

If

x = 0

, then

y = c

, which gives us the regression point

A(0, c)

step 12

If

ax + b = 0

, then

x = -\frac{b}{a}

and

y = a\left(-\frac{b}{a}\right)^2 + b\left(-\frac{b}{a}\right) + c

step 13

Simplifying

y

for

x = -\frac{b}{a}

gives us

y = \frac{b^2}{a} - \frac{b^2}{a} + c = c

step 14

Therefore, the regression point

A

is

\left(-\frac{b}{a}, c\right)

or

(0, c)

Answer

The regression points on the parabola are

A\left(-\frac{b}{a}, c\right)

and

A(0, c)

.

Key Concept

Symmetry in Parabolas

Explanation

A regression parabola is symmetric with respect to the origin, meaning that if a point

A

is on the parabola, its reflection

A'

across the origin is also on the parabola. The regression points are found by setting the original and reflected points equal to each other in the parabola's equation.

鸡兔同笼现在有脚80有头30问鸡兔各多少

Solution by Steps

step 1

Translate the problem from Chinese to English: "Chicken and rabbit in a cage, there are 80 feet and 30 heads in total. How many chickens and rabbits are there?"

step 2

Let

c

represent the number of chickens and

r

represent the number of rabbits. Chickens have 2 feet and rabbits have 4 feet

step 3

Set up the system of equations based on the information given:

2c + 4r = 80

(total feet) and

c + r = 30

(total heads)

step 4

Multiply the second equation by 2 to align with the number of feet:

2c + 2r = 60

step 5

Subtract the modified second equation from the first equation to solve for

r

:

(2c + 4r) - (2c + 2r) = 80 - 60

step 6

Simplify to find

r

:

2r = 20

then

r = 10

step 7

Substitute

r = 10

into the second original equation to find

c

:

c + 10 = 30

step 8

Solve for

c

:

c = 20

Answer

There are 20 chickens and 10 rabbits.

Key Concept

System of Linear Equations

Explanation

The problem is a classic example of a system of linear equations where the number of variables equals the number of unique equations, allowing for a unique solution.

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