Objective Introduce the Quadratic Formula and learn how to use it to solve quadratic equations.
In this lesson, you will use the Quadratic Formula to solve quadratic equations. The goal is that you become comfortable using this formula. This will require lots of practice, and so you should pay special attention to the examples.
The Quadratic Formula The solutions of a quadratic equation in the form ax^{ 2} + bx + c = 0, where a 0, are given by the formula .
This formula can be used to find the solutions to any quadratic equation.
Notice the sign in the formula. This means that there may be two solutions to a quadratic equation. Recall that a parabola intersects the x -axis in two places, the corresponding equation has two solutions. This corresponds to the two solutions .
The expression inside of the square root, b^{ 2} - 4ac, is called the discriminant. When the discriminant is positive, there are two distinct solutions. When this expression is zero, the square root in the Quadratic Formula is zero, and then there is only one solution, . This corresponds to the parabola intersecting the x -axis in only one point, namely the vertex. The final case is when the discriminant b^{ 2} - 4ac is negative, in which case the square root is not defined. In this case, there are no real solutions to the quadratic equation. This corresponds to the parabola not intersecting the x-axis at all. A variety of algebra solvers can help you solving problems like this one
Solutions of the Quadratic Equation ax^{ 2} + bx + c = 0 | |||
Discriminant | b^{ 2} - 4ac > 0 | b^{ 2} - 4ac = 0 | b^{ 2} - 4ac < 0 |
Number of Solutions | 2 |
1 |
0 |
Example | |||
Parabola Intersects the x-axis | yes, in two distinct points | yes, in exactly one point, the vertex | no |
Algebra equations:An arithmetic or algebra equation is simply an arithmetic or algebra expression with an = sign. For example the arithmetic equations 3 + 2 = 5 or 3 * 5 = 15 and the algebra equations 3 + x =5 or 3 * x = 15. Quadratic equation:in mathematics, a polynomial equation of second degree (that is, an equation containing as its highest power the square of a variable, such as x2). The general formula of such equations is ax2 + bx + c = 0, in which a, b, and c are real numbers, and only the coefficient a cannot equal 0. In coordinate geometry, a quadratic function represents a parabola. Simplifying:What is it? Reducing to lowest terms. Radical:The square root sign. Algebra:What is it? A branch of mathematics in which variables are substituted for unknown values to solve a particular problem. Quadratic equation:An equation of degree 2, which has at most two solutions. Substitution:1. n. as defined below. 2. n. a method of solving a system of equations by replacing one variable with an equivalent expression in the other variable. Math:Mathematics is the study of quantity, structure, space and change. It developed, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. Radical expression:an expression especially an irrational number or quantity involving a radical sign. Calculator:A machine used for doing mathematical calculations. Factorising:Finding what to multiply to get an expression. (Called Factoring in US English.) Example: 2y+6 = 2(y+3), so the factors of 2y+6 are 2 and (y+3) Logarithm:The exponent of the power to which a base number must be raised to equal a given number. Example: 2 is the logarithm of 100 to the base 10 (2=log10100). (10 must be raised to the power of 2 in order to equal 100) Quadratic equation:An equation involving the second power, but no higher power of an unknown. The general form of a quadratic equation in two unknowns is: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. Grade:The slope of a road, often represented as a percent or ratio. Algebra:The mathematics of working with variables. Linear equation:- Ax + By + C = 0 Math:Mathematics is the study of quantity, structure, space and change. It developed, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. Coordinate graph:graphical representation of pairs of related numerical values. The data are sorted into pairs of numbers with each pair associated with one person (for example, height and armspan of each person measured). Algebra:Algebra is a continuation and extension of the rules of arithmetic into a more general level. Average:see: Mean (or average) Algebraic expression:expressions composed of letters to stand for numbers. Substitution method:Substitution method is a method of solving a system of equations wherein one of the equations is solved for one varaible in terms of the other variables. Algebra:Algebra is the study of generalized arithmetic. In algebra, unknown numbers can be represented by letters in order to solve equations. For example, 4 + x = 10 is true for x=6. Algebra (originally called al-jabr, meaning "restoration") was invented in the Middle East by Abu Ja'far Muhammad ibn Musa Al-Khwarizmi (born in Baghdad about AD 825) during the Middle Ages. Calculator:n. a machine used for computation possibly having computer ability. Equation:An algebraic expression asserting the equality of two quantities.Example: x + 4 = y -10 Square root:Represented by this symbol ?? a square root of any particular number is a number, which when multiplied by itself, will produce the given number. NB as two negative numbers multiplied together will result in a positive answer it is important to realise that square roots can be positive or negative numbers. Thus the square root of 9 (??9) is either +3 or -3. Calculator:A machine for performing arithemtical calculations. Fraction:A part of a whole number. Division:Division is an operation that tells us the number of groups that can be made out of a number of items or how many items will be in a group. The symbol ?? denotes division. Algebra:A type of math in which variables (letters that represent numbers) are combined according to basic arithmetic rules. Function:A function is a rule connecting two sets such that for each item in the first set there is just one item which it is related to in the second set. The mathematical concept of a function expresses dependence between two quantities, one of which is given and the other produced. A function associates a single output to each input element drawn from a fixed set, such as the real numbers. Tto specify a function the input set (the domain) and the output set (the range) should be stated. Usually they |