Algebraic Fractions

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Algebraic Fractions What is an algebraic fraction? An algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. We will explain some techniques to simplify complex algebraic fractions. We know that we can simplify fractions by dividing the numerator and denominator by a common factor. This can also be done with algebraic fractions. Example […]

Linear Inequalities

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Linear inequalities Number lines Suppose that you are told that x<4. This means that each possible value of x must be less than 4. Therefore, x can be 3, 2, 1, 0, -1, -2…….but that is not all. 3.2 is also less than 4, as is 3.999, 2.43, -3.4, -100…….. If we draw a number […]

Rearrangement of a Formula

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Rearrengement of a formula Very often you will find that a formula is expressed with one variable written alone in one side of the ‘=’ symbol (usually on the left but not always). The variable that is written alone is known as the subject of the formula. Consider each of the following formulas $ displaystyle s=ut+frac{1}{2}a{{t}^{2}}$   ($ […]

Dividing Polynomial Expressions

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Dividing Polynomial Expressions What is a Polynomial? A polynomial is an expression obtained when we add or subtract two or more monomials. Dividing Polynomials Dividing polynomials with monomial expression is easy, all you have to do is divide each term of the polynomial by the monomial. Example 1: Dividing $displaystyle {10x+5}$ by $displaystyle 5$. $displaystyle frac{{10x+5}}{5}=frac{{10x}}{5}+frac{5}{5}=2x+1$ Example 2 :Dividing $displaystyle frac{{12{{x}^{2}}+8x+20}}{{4x}}=$ […]

Rules of Roots

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Rules of Roots What is the nth root? The number that must be multiplied itself n times to equal a given value. The nth root of x is written $ displaystyle sqrt[n]{x}$ or $ displaystyle {{x}^{{frac{1}{n}}}}$. The rules below are a subset of the rules of exponents, because roots are the inverse operations of exponentiation. Definitions […]

Rules of Exponents

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Rules of Exponents Zero-Exponent Rule:   $displaystyle {{a}^{0}}=1$  this means that each number raised to the zero power is 1. Worked example 1:  50 = 1 (-3)0 = 1 $displaystyle {{(5{{x}^{6}}{{y}^{2}})}^{0}}=1$ $displaystyle {{left( frac{2}{x} right)}^{0}}=1$ Power of a Power: $displaystyle {{left( {{a}^{m}} right)}^{n}}={{a}^{mn}}$ So, when you raise a power to a power you have to multiply the exponents. Worked example 2: […]

Logarithm

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Logarithm Logarithm PropertiesThere is a close relationship between logarithms and exponents. The logarithm of a number is defined as the power or index to which a given base must be raised to obtain the number. If we have $ displaystyle {{a}^{x}}=M$ where a  and M are greater than zero and a≠1 then we can write this in a logarithm […]

Worked Examples – Quadratic Equations

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Worked Examples – Quadratic equation Methods to solve a quadratic equation are: Square Root Method Quadratic FormulaFactoring Completing the square Graphing We have four types of quadratic equations 1. Quadratic equations in the standard form $ displaystyle a{{x}^{2}}+bx+c=0$ We can solve them by using the quadratic formula, graphing or factoring and completing the square. 2. Quadratic […]

Combining Functions

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Combining Functions Combining Functions means performing basic arithmetic operations like addition, subtraction, multiplication and division with functions. Given two functions $ displaystyle f(x)$ and $ displaystyle g(x)$ we define: 1. The sum of two functions $displaystyle (f+g)(x)=f(x)+g(x)$ 2. The difference of two functions $ displaystyle (f-g)(x)=f(x)-g(x)$ 3. The product of two functions $ displaystyle (ftimes […]

Worked examples – First degree equations

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Worked Examples – First degree equations Based on what we have explained to the article First degree Equations, we are going to solve some equations below: Example 1 Solve the equation: a) $ displaystyle 5x=-60$$ displaystyle x=frac{{-60}}{5}$$ displaystyle x=-12$ b) $ displaystyle -3x=-9$$ displaystyle x=frac{{-9}}{{-3}}$$ displaystyle x=3$ c) $ displaystyle frac{1}{3}x=12$$ displaystyle x=frac{{12}}{{frac{1}{3}}}=12cdot 3$$ displaystyle x=36$ d) $ displaystyle x=-x$$ […]

Equations and Gradients of a straight line

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Equations and Gradients of a straight line What is a straight line? A straight line is a line with no curves. By definition we can say that a straight line is the set of all points between and extending beyond two points. Equation of a straight line An equation of the first degree of the […]

Expanding and Factorising Algebraic Expressions

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Expanding and Factorising Algebraic expressions What is an expansion? The process of re writing an algebraic expression that contains brackets by expanding them is called an expansion.To expand a bracket means to multiply each term in the bracket by the expression outside the bracket. Be careful! Watch out for negative numbers in front of brackets […]

Algebraic Expressions

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Algebraic expressions What is an algebraic expression? An algebraic expression is an expression which is built from variables, constants and the algebraic operations. Difference between algebraic expression and algebraic equation An algebraic expression is a mathematical phase that can contain numbers, variables and not an equal sing .They can´t be solved but they can be […]

Simultaneous linear equations

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Simultaneous linear equations What are simultaneous equations? Simultaneous equations are a set of two or more equations, each containing two or more variables whose values can simultaneously satisfy both or all the equations in the set, the number of variables being equal to or less than the number of equations in the set. Simultaneous linear […]

Quadratic Equation

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Quadratic equation What is a quadratic equation? A quadratic equation is an equation of degree 2 with one variable and that can be written as: x² + bx + c = 0, where a≠0, a, b, c are the coefficients and x is the variable. The number of roots of a quadratic equation is less or equal to the number of its degree. […]

Equations. First degree equation

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First degree equation What is an equation? An equation in mathematics is a statement that links 2 expressions with the equal symbol. One of the basic type of equations are the first degree equations,that can be called as ‘’Linear equations”. Linear Equations with one variable A linear equation is an equation that represents a line on […]

Powers and Roots

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Powers and Roots An expression that represents repeated multiplication of the same factor is called a power. So, when we want to multiply a number by itself repeatedly we use powers. $ displaystyle {{a}^{n}}$ tells  you  to multiply  a  by itself, exactly n-times. The exponent corresponds to the number of times the base is used as […]