Quadratics in One Variable
Quadratic Equation
A QUADRATIC EQUATION is any equation of the form equation: ax + bx + c = 0, where a ≠ 0 A quadratic equation usually is solved in one of four algebraic ways: [1] Factoring [2] Applying the square root property [3] Completing the square [4] Applying the quadratic formula
Solving Radical Equations
A RADICAL EQUATION is an equation in which a variable is under a radical. To solve a radical equation: [1] First, isolate the radical expression involving the variable. If more than one radical expression involves the variable, then isolate one of them. [2] Next, raise both sides of the equation to the index of the radical. [3] If there is still a radical equation, repeat steps 1 and 2; otherwise, solve the resulting equation and check the answer in the original equation. By raising both sides of an equation to a power, some solutions may have been introduced that do not make the original equation true. These solutions are called EXTRANEOUS SOLUTIONS. Copy and paste the following link into your browser to learn more about solving radical equations: https://youtu.be/kscLqC-zmBg
Solving Equations in Quadratic Form
Any equation in the form ax 2 + bx + c = 0 is said to be in QUADRATIC FORM. This equation then can be solved by using the quadratic formula, by completing the square, or by factoring if it is factorable. Copy and paste the following link into your browser to learn more about solving equations in the quadratic form: https://youtu.be/pzRqipVKHd4
Solving Quadratric Inequalitities
QUADRATIC INEQUALITY is an inequality in which one side is a quadratic polynomial and the other side is zero. To solve a quadratic inequality, follow these steps: [1] Approach the solution for an inequality as though it were an equation. The real solutions to the equation become boundary points for the solution to the inequality. [2] Make the boundary points solid circles if the original inequality includes equality; otherwise, make the boundary points open circles. [3] Select points from each of the regions created by the boundary points. Replace these "test points" in the original inequality. [4] If a test point satisfies the original inequality, then the region that contains that test point is part of the solution. [5] Represent the solution in graphic form and in solution set form. Copy and paste the following link into your browser to learn more about solving quadratic inequalities: http://www.bing.com/videos/search?q=Solving+Quadratric+Inequalitities&&view=detail&mid=AAFA785BF4E1DA99E6D5AAFA785BF4E1DA99E6D5&FORM=VRDGAR
Solving Quadratics by Factoring
SOLVING A QUADRATIC EQUATION BY FACTORING depends on the zero product property. The zero product property states that if ab = 0, then either a = 0 or b = 0. For example: Solve 2x² = -9x - 4 by using factoring. [1] First, get all terms on one side of the equation, by adding '9x + 4' to both sides: 2x² + 9x + 4 = 0 [2] Factor the quadratic: (2x + 1)(x + 4) = 0 [3] Apply the zero product property: 2x + 1 = 0, then x = -½ ****OR**** x + 4 = 0, then x = -4 REMEMBER THE ZERO PRODUCT PROPERTY SPECIFIES: In algebra, the zero-product property states that the product of two non-zero elements is non-zero. The zero-product property is also known as the rule of zero product, the null factor law or the nonexistence of nontrivial zero divisors. Copy and paste the following link into your browser to learn more about solving quadratics by factoring: http://www.bing.com/videos/search?q=Solving+Quadratics+by+Factoring&&view=detail&mid=7EAF76F1B348DF937E3C7EAF76F1B348DF937E3C&FORM=VRDGAR
Solving Quadratics by Quadratic Formula
The QUADRATIC FORMULA uses the "a", "b", and "c" from "ax² + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients" of the quadratic equation they've given you to solve. The Quadratic Formula is derived from the process of completing the square, and is formally stated as: x = -b ± √b² - 4ac / 2a ☛☛☛ REMEMBER: This Quadratic Formula can be used to solve ALL quadratic equations. Copy and paste the following link into your browser to learn more about solving quadratics by using the quadratic formula: http://www.bing.com/videos/search?q=solving+quadratics+by+using+the+quadratic+formula&&view=detail&mid=F3CD6BF5AA3AB4B49329F3CD6BF5AA3AB4B49329&FORM=VRDGAR
Solving Quadratics by the Square Root Property
The SQUARE ROOT PROPERTY says that if x² = c, then x = √c, or x = -√c. This can be written as "if x² = c, then equation." If c is positive, then x has two real answers. If c is negative, then x has two imaginary answers. Copy and paste the following link into your browser to learn more about solving quadratics by the square root property: http://www.bing.com/videos/search?q=Solving+Quadratics+by+the+Square+Root+Property&&view=detail&mid=43CCBF54F0BAC321ACEA43CCBF54F0BAC321ACEA&FORM=VRDGAR
Solving Quadratics by Completing the Square
The expression x² + bx can be made into a square trinomial by adding to it a certain value. This value is found by performing two steps: [1] First, multiply 'b' (the coefficient of the "x‐term") by ½. [2] Then SQUARE THE RESULT. For example: Find the value to add to x² + 8 x to make it become a square trinomial: ☛ Multiply the coefficient of the "x‐term" by ½, which is 8 • ½ = 4 ☛ Square that result., which is (4)² = 16 So, 16 must be added to x² + 8x to make it a square trinomial: x² + 8x + 16 = (x + 4)² Finding the value that makes a quadratic become a square trinomial is called COMPLETING THE SQUARE. That square trinomial then can be solved easily by factoring. To solve quadratic equations by using the completing the square method, the coefficient of the squared term must be 1. If it isn't, then first divide both sides of the equation by that coefficient and then proceed as before. Copy and paste the following link into your browser to learn more about solving quadratics by completing the square: http://www.bing.com/videos/search?q=Solving+Quadratics+by+Completing+the+Square&&view=detail&mid=0B3DEACCD690B9178A5F0B3DEACCD690B9178A5F&FORM=VRDGAR
