Algebra

28.45 or28.:4 or28.430

To find the number of students who don't take either subject, we first calculate the total number of students taking at least one subject using the formula for the union of two sets:

Total taking at least one = (Algebra 1) + (Algebra 2) - (Both) = 14 + 20 - 7 = 27.

Therefore, the number of students not taking either subject is 60 - 27 = 33.

The formula for the standard deviation of a proportion ( p ) in a sample is given by ( \sqrt{\frac{pq}{n}} ), where ( p ) is the proportion of successes, ( q ) is the proportion of failures (i.e., ( q = 1 - p )), and ( n ) is the sample size. This formula applies when the sample size is sufficiently large and the expected number of successes and failures (i.e., ( np ) and ( nq )) are both greater than 5, ensuring that the normal approximation is valid. It is used to estimate the standard deviation of the sampling distribution of the sample proportion.

NO!!

When multiplying/dividing

• X - X - X = + X - = -

Note ; when you mukltiply/divide the first pair of negative number, they produce a positive answer.

To this positive answer you then multiply another negative number giving a negative answer.

127

When the result of solving a linear equation is ( x = 0 ), it means that the value of the variable ( x ) that satisfies the equation is zero. This indicates that the equation holds true when ( x ) is substituted with 0. In a graphical context, this result implies that the line represented by the equation intersects the x-axis at the origin (0,0).

A sixth-degree polynomial function can have up to six unique roots. However, the actual number of unique roots can be fewer than six, depending on the specific polynomial and whether some roots are repeated (multiplicity). According to the Fundamental Theorem of Algebra, the total number of roots, counting multiplicities, will always equal the degree of the polynomial, which is six in this case.

Area = pi radius squared

Algebraically

A = pi r^(2)

The radius is one half the diiameter . Since the diameter is 12 ft , then the radius is 6 ft .

pi = 3.141592..... (a constant)

Substituting #

A = pi (6ft)^(2)

A = 3.141952... 36 sq.ft.

A = 113.097.... sq. ft.

The number 0.000041 in scientific notation is expressed as 4.1 × 10⁻⁵. This format highlights the significant figures and indicates the decimal point has moved five places to the right.

Combinations of numbers and operations refer to the various ways in which numerical values can be combined using mathematical operations such as addition, subtraction, multiplication, and division. These combinations can create expressions or equations, allowing for the manipulation and analysis of numerical relationships. For example, the combination of the numbers 3 and 5 with the operation of addition results in the expression 3 + 5, which equals 8. Such combinations are fundamental to arithmetic and form the basis for more complex mathematical concepts.

The 9x table is a multiplication table that lists the products of the number 9 multiplied by integers from 1 to 10 (and beyond). It includes values like 9 (9x1), 18 (9x2), 27 (9x3), and so on, up to 90 (9x10). A notable pattern in the 9x table is that the digits of each product add up to 9, for example, 18 (1+8=9) and 27 (2+7=9). This table is often used in elementary math to help students learn multiplication.

r^(4) / r^(6) = r^(4-6) = r^(-2) = 1/r^(2)

Rules for manipulation of indicies.

#1 ; the coefficient MUST be the same . 'r' in this case

#2 ; for multiplication ; ; add the indices

#3 ; for division ; subtract the indices

#4 ' for 'nesting' ; multiply the indices

Careful something of the nature 'r^(2)' X s^(3)' CANNOT be done as the coefficients ''r' & 's' are different. The coefficients MUST be the same

10^(6) =

1,000,000 in word 'one million'.

5^(-6) =

1/ (5^(6)) =

1/15625 ( some 'horribly' small fraction). !!!!!

'Five to Six' ??? There are two points in the day , when this time is recorded.

In the morning it is 5:55 am , that is it is five minutes to six o'clock.

Similarly in the evening it is 5:55 pm that is it is five minutes to six o'clock.

'am/pm' are the initials of the Latin phrase ' ante/post meridian ' meaning ' before/after noon'.

'Six o' clock' is a corruption of the English language for 'six hours of the clock'.

Originally, analogue clocks were made to show only 12 hours at a time.

On an analogue clock 'six o'clock' would have the large finger/hand pointing to '12'. and the small finger/hand pointing to '6'.

At 'five to six', the large finger/hand would point to '11' meaning that there are 5 minutes to the 'hour(12) of the 'clock'. The small finger/hand would point almost to '6'.

On the analogue clock , between each number , there are five small divisions, that indicate minutes. There are 60 of these divisions in one full dial of the clock-face. The numbers refer to the hour of time. The numbers can be in Arabic(modern), Roman or Symbol form.

To solve the expression involving square roots of the form (\sqrt{a + bx} + \sqrt{b + cx} + \sqrt{c + ax}), you can analyze it by substituting specific values for (x) or using calculus to find critical points. Additionally, applying algebraic manipulation or inequalities (like the Cauchy-Schwarz inequality) could help in simplifying the expression. If you're looking for a maximum or minimum value, consider differentiating the expression with respect to (x) and solving for critical points.

Numerical expressions are used to represent quantitative relationships and calculations in real-world problems. By translating scenarios into mathematical terms, we can perform operations like addition, subtraction, multiplication, and division to find solutions. For instance, if you're budgeting for groceries, you can use a numerical expression to sum the costs of each item to ensure you stay within your budget. This approach allows for clear, logical problem-solving and decision-making in various contexts.

As of October 2023, the Chicago Bears have a record of 1 win and 5 losses in the current NFL season. Their performance has been challenging, contributing to their position in the standings. For the most accurate and up-to-date statistics, it's best to check the latest sports news or the NFL's official website.

The problem-solving process typically involves several key steps:

1. Identifying the Problem: Clearly define the issue at hand to understand its nature and scope.
2. Gathering Information: Collect relevant data and insights that can inform potential solutions.
3. Generating Solutions: Brainstorm possible solutions, considering various approaches and their feasibility.
4. Implementing and Evaluating: Choose the best solution, implement it, and assess its effectiveness, making adjustments as necessary.

The equation of a hyperbola with a transverse axis of length 18 can be expressed in the standard form (\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1) or (\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1), depending on its orientation. Here, (a) is half the length of the transverse axis; thus, (a = 9). The equation would then be (\frac{(y-k)^2}{9^2} - \frac{(x-h)^2}{b^2} = 1) for a vertical hyperbola, or (\frac{(x-h)^2}{9^2} - \frac{(y-k)^2}{b^2} = 1) for a horizontal hyperbola, where ((h, k)) is the center of the hyperbola.

7x^(2)

= 7

Divide both sides by '7'

Hence

x^(2) = 1

Square root both sides

x = (+)1 & (-)1 , usually written as ' +/- 1 '.

Remember

(+) X (+)1 = 1

&

-1 X -1 = 1

7 x a is written as ' 7a '

In algebra Plus is shown as ' a + b '

Minus is shown as ' a - b'

Multiplication is shown as 'ab' . NEVER a x b , as the 'x' may be confused with an unknown value.

Division is shown as ' a/b ' , like a fraction.

e.g.

6a^(2) + 5b - 2c/3 means

Six multiplied to 'x' and them multiplied to 'x' again (squared). Then you 'add' five multiplied to 'b' , then you subtract two multiplied to 'c' and the whole '2c' is divided by '3'.

The divisional three only refers to the '2c'. If you wanted to divided the whole expression by '3' , then you insert brackets.

[ 6a^(2) + 5b - 2c ] /3

Note the position(s) of the 'square brackets'.

To find the roots of the polynomial ( x^2 - 11x + 15 ), we can factor it. The polynomial factors to ( (x - 5)(x - 3) = 0 ). Therefore, the two values of ( x ) that are roots of the polynomial are ( x = 5 ) and ( x = 3 ).

To find the first four terms of the sequence defined by ( a_n = 2n - 8 ), we can substitute ( n = 1, 2, 3, ) and ( 4 ).

1. For ( n = 1 ): ( a_1 = 2(1) - 8 = 2 - 8 = -6 )
2. For ( n = 2 ): ( a_2 = 2(2) - 8 = 4 - 8 = -4 )
3. For ( n = 3 ): ( a_3 = 2(3) - 8 = 6 - 8 = -2 )
4. For ( n = 4 ): ( a_4 = 2(4) - 8 = 8 - 8 = 0 )

Thus, the first four terms of the sequence are (-6, -4, -2, 0).

To simplify the expression ( 2x - 2y + 5z - 2x - y + 3z ), combine like terms. The ( 2x ) and ( -2x ) cancel each other out, leaving ( -2y - y = -3y ) and ( 5z + 3z = 8z ). Thus, the simplified expression is ( -3y + 8z ).