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Apuntes gratuitos de Matemáticas
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Apuntes gratuitos de Matemáticas
16 apuntes
How do you solve a quadratic equation using the quadratic formula?
Write the equation as ax² + bx + c = 0, read off a, b and c, then substitute them into x = (−b ± √(b² − 4ac)) / 2a. Work out the discriminant b² − 4ac first: if it is…
How do you find the gradient of a straight line from two points?
The gradient is the change in y divided by the change in x. For points (x₁, y₁) and (x₂, y₂), gradient m = (y₂ − y₁) / (x₂ − x₁). A positive gradient slopes upwards from…
How do you expand and simplify double brackets like (x + 3)(x − 5)?
Multiply every term in the first bracket by every term in the second, then collect like terms. For (x + 3)(x − 5): x × x = x², x × (−5) = −5x, 3 × x = 3x, 3 × (−5) =…
How do you differentiate a polynomial?
Differentiate each term separately using the power rule: for axⁿ, multiply by the power and reduce the power by one, giving naxⁿ⁻¹. A term in x becomes a constant and a…
How do you calculate a percentage increase?
Percentage increase = (new value − original value) ÷ original value × 100. For example, a price rising from 80 to 100 is an increase of 20, and 20 ÷ 80 × 100 = 25%…
What is Pythagoras' theorem and how do you use it?
In a right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c²…
How do you add fractions with different denominators?
Make the denominators the same first. Find a common denominator (the lowest common multiple is easiest), change each fraction to an equivalent fraction with that…
How do you solve simultaneous equations by elimination?
Make the coefficient of one letter the same in both equations, then add or subtract the equations to remove that letter. Solve for the letter that is left, then…
How do you simplify surds?
To simplify a surd, write the number under the root as a product that includes the largest possible square number, then take the square root of that square outside. For…
How do you solve a linear equation?
Get the unknown on its own by doing the same operation to both sides of the equation. Undo additions and subtractions first, then multiplications and divisions. For 3x +…
How do you find the area and circumference of a circle?
Area of a circle = πr², where r is the radius. Circumference = 2πr, or πd where d is the diameter. Always check whether you are given the radius or the diameter: the…
How do you calculate probability?
Probability = number of favourable outcomes ÷ total number of equally likely outcomes. It is always between 0 (impossible) and 1 (certain), and can be written as a…
How do you find the mean, median, mode and range?
The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the most common value. The range is the…
How do you use SOHCAHTOA to find a side or angle?
In a right-angled triangle, sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse and tan θ = opposite ÷ adjacent. Label the sides relative to the angle, pick the…
What are the laws of logarithms?
The three main laws are: log a + log b = log(ab), log a − log b = log(a ÷ b), and log(aⁿ) = n log a. They hold for any base as long as both logs use the same base. A…
How do you integrate a polynomial?
Integrate each term separately: for axⁿ, add one to the power and divide by the new power, giving axⁿ⁺¹/(n + 1). Always add a constant of integration, + c, for an…