Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers, quantities, and operations. It is a unifying thread of almost all mathematics and deals with equations, functions, and structures. In grade 12 algebra, you will explore complex concepts that build on previous algebraic principles. This comprehensive guide delves deep into the basic ideas and examples to enhance your understanding of algebra.

Basic concepts of algebra

In algebra, letters such as x, y, and z are often used to represent variables. These variables can take different values. Algebraic expressions are created by combining variables and numbers with mathematical operations. An important part of algebra involves simplifying these expressions, solving equations, and understanding functions.

Algebraic expression

An algebraic expression is a combination of numbers, variables, and operations. For example, 3x + 2 is an algebraic expression where 3 is a coefficient, x is a variable, and 2 is a constant.

Example

Simplify the following expression: 2x + 3x - 5 + 4.

Step 1: Combine like terms: 2x and 3x are like terms.
2x + 3x = 5x

Step 2: Combine the constants: -5 and 4.
-5 + 4 = -1

Simplified expression: 5x - 1

Equation

An equation is a mathematical statement that asserts the equality of two expressions. For example, in the equation 3x + 2 = 11, our job is to find the value of x that makes the equation true.

Solving linear equations

Linear equations are algebraic equations of the form ax + b = 0, where a and b are constants.

Example

Solve the equation: 3x + 2 = 11.

Step 1: Subtract 2 from both sides to isolate 3x.
3x + 2 - 2 = 11 - 2
3x = 9

Step 2: Divide both sides by 3 to find x value.
3x / 3 = 9 / 3

x = 3

Quadratic equations

Quadratic equations are polynomials of degree two, usually in the form ax^2 + bx + c = 0.

There are several ways to solve quadratic equations:

• Factoring
• Completing the square
• Quadratic formula

Factoring

Factoring involves writing the quadratic expression as the product of two binomials.

Example

Solve x^2 - 5x + 6 = 0 by factoring.

Step 1: Factor out the quadratic expression: Look at the two numbers that multiply by 6 and add up to -5.

(x - 2)(x - 3) = 0

Step 2: Put each factor equal to zero and solve for x.
x - 2 = 0 or x - 3 = 0

Solution: x = 2 or x = 3

Completing the square

Completing the square involves converting a quadratic equation into a perfect square trinomial.

Example

Solve x^2 + 6x + 5 = 0 by completing the square.

Step 1: Move the constant to the other side.
x^2 + 6x = -5

Step 2: Take half of the coefficient of x, square it, and add it to both sides.
(6/2)^2 = 9
x^2 + 6x + 9 = 4

Step 3: Write as a perfect square.
(x + 3)^2 = 4

Step 4: Find the square root of both sides.
x + 3 = ±√4

Step 5: Solve for x.
x = -3 ± 2

Solution: x = -1 or x = -5

Quadratic formula

The quadratic formula provides a reliable way to solve any quadratic equation: x = (-b ± √(b^2 - 4ac)) / (2a).

Example

Solve 2x^2 - 4x - 6 = 0 using the quadratic formula.

Step 1: Identify a, b and c.
a = 2, b = -4, c = -6

Step 2: Plug in the quadratic formula.
x = (-(-4) ± √((-4)^2 - 4*2*(-6))) / (2*2)
x = (4 ± √(16 + 48)) / 4
x = (4 ± √64) / 4

Step 3: Solve for x.
x = (4 ± 8) / 4

Solution: x = 3 or x = -0.5

Work

A function is a special relationship where each input has a single output. Functions can be expressed in a variety of forms, including equations, graphs, and tables.

Example: linear function

A linear function represents a straight-line relationship between variables and is usually written as f(x) = mx + b, where m is the slope and b is the y-intercept.

Graphical representation:

Example

Given f(x) = 2x - 5 Find the value for x = 3.

f(3) = 2(3) - 5 = 6 - 5 = 1

Thus, f(3) = 1.

Quadratic functions

A quadratic function is characterized by a parabolic graph and can be expressed as f(x) = ax^2 + bx + c.

Graphical representation:

Example

Given f(x) = x^2 - 4x + 3 Evaluate for x = 2.

f(2) = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1

Thus, f(2) = -1.

Polynomial function

Polynomials are algebraic expressions consisting of a sum of powers multiplied by coefficients in one or more variables. The general form of a polynomial function in one variable is: f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0.

The degree of a polynomial is the highest power of the variable. For example, in f(x) = 4x^3 + 3x^2 + 2x + 1, the degree is 3.

Example

Identify the degree of the following polynomial: 7x^4 + x^3 - 3.

The degree is 4.

Rational expressions

Rational expressions are fractions whose numerator and/or denominator are polynomials. Simplifying rational expressions involves factoring and subtraction.

Example

Simplify the rational expression: (x^2 - 9)/(x^2 - 3x).

Step 1: Factor the numerator and denominator.
Fraction: x^2 - 9 = (x + 3)(x - 3)
Denominator: x^2 - 3x = x(x - 3)

Step 2: Cancel out the common factors.

((x + 3)(x - 3))/(x(x - 3)) = (x + 3)/x

Exponential and logarithmic functions

Exponential and logarithmic functions are advanced algebraic concepts. The exponential function is f(x) = a^x, where a is a constant. The logarithmic function is the inverse: f(x) = log_a(x).

Properties and laws

• Product Rule: log_a(xy) = log_a(x) + log_a(y)
• Quotient Rule: log_a(x/y) = log_a(x) - log_a(y)
• Power Rule: log_a(x^k) = k log_a(x)

Example

Find the solution for x: 3^x = 81.

Step 1: Express 81 as a power of 3.
81 = 3^4

Step 2: Since the bases are the same, keep the exponents the same.

x = 4

Conclusion

Algebra at the grade 12 level involves understanding and solving complex equations, functions, and expressions. It extends the basic concepts, introducing new techniques and ideas to deal with polynomial, rational, exponential, and logarithmic functions. Mastering these concepts is important, as they form the basis for advanced studies in mathematics and applied sciences.

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