Solution of Equations

Solution of Equations

Types of Equations

• Linear Equations: These are of the form ax + b = 0, where a and b are constants and x is a variable. The solution is given by x = -b/a.
• Quadratic Equations: Equations of the form ax^2 + bx + c = 0. The solutions are given by the quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / 2a.
• Simultaneous Equations: A system of equations that are solved together. Linear simultaneous equations can be solved using substitution or elimination.
• Polynomial Equations: These are equations that can be expressed as a polynomial equal to zero. Various techniques can be used to solve these, including long division and synthetic division.
• Rational Equations: Equations that contain a rational expression, which is an expression in the form of a fraction where the numerator and/or the denominator are polynomials.
• Radical Equations: Equations involving roots. To solve these, the square (or any other root) is often isolated on one side, and both sides of the equation are squared to remove the root.

Techniques to Solve Equations

• Factorisation: For an equation to be solved by factorisation, one side must be put in the form of a factorised polynomial equal to zero. Each factor is then set equal to zero, yielding the solutions.
• Completing the Square: A method used to solve difficult quadratics by rearranging the quadratic to the form (x-h)^2 = k. Square rooting both sides gives solutions.
• Using the Quadratic Formula: Employed when an equation is difficult to factorise or when completing the square is tedious.
• Substitution: Used generally when solving simultaneous equations or equations with more than one variable. It involves replacing a variable in an equation with an expression from another equation.
• Elimination : Another method for solving simultaneous equations by making the coefficients of one of the variables the same, then subtracting one equation from the other.
• Rearranging: This might involve multiplying or dividing both sides of the equation, using the distributive property or combining like terms. The aim is to isolate the variable on one side of the equation
• Cross Multiplication: A method used to solve equations where a fraction equals a fraction.

Checking your solution

• Substitute back into the equation: After solving an equation, it is important to check your solution by substitifying it back into the original equation. This will ensure the solutions are correct.

Solving Inequalities

• How to solve: Inequality rules are similar to those of equations, with one key difference that the direction of the inequality changes when both sides are multiplicated or divided by a negative number.
• Interval notation: The solution of an inequality is often expressed as an interval or a combination of intervals. It is important to accurately represent open and closed intervals (whether the end-points are included).