Solving Quadratic Equations

Quadratic equations are equations where the highest power is 2. Quadratic equations look like this: ax ² + bx + c. Sometimes the b value is missing, other times the c value is missing. Quadratic equations can be solved two different ways by factoring or using the quadratic formula. Solving a quadratic equation means finding the  roots of the equation. Roots are the solutions which may satisfy the equation. 

Factoring:
Factoring a quadratic equation is only possible if the equation is factor-able. For example: 

Here the equation is x ² + x - 42. It can be factored to (x+7)(x-6). Than you set them to zero and solve for x. The solutions for x are -7 and 6. 

The Quadratic Formula:
The quadratic formula works for every quadratic equation even the ones that can be factored. The quadratic formula is  . In order to solve the equation using the formula you must substitute the numbers in the equation. For example:

Here the equation is x ² + 2x - 3. The a is 1, b is 2 and c is -3. Here all the numbers are plugged in and solved for. Reminder, always find both solutions of the equation by subtracting and adding the numbers properly using the order of operations. The roots for this solution is 1 and -3.

Question:
What are the roots for this quadratic equation? x² + x – 4 = 0





Sources:

• http://www.google.com/url?source=imglanding&ct=img&q=http://www.gradeamathhelp.com/image-files/solve-by-factoring.jpg&sa=X&ei=pWuMUOOSNcr20gHqpID4Dw&ved=0CAkQ8wc&usg=AFQjCNE7eD9GbzXwexGHnCXHnZOttkkdnw
• http://www.google.com/url?source=imglanding&ct=img&q=http://www.mathwarehouse.com/quadratic/images/formula_solution_-3_and_1.gif&sa=X&ei=Bm-MUPCmD4f50gHJtoHgCg&ved=0CAkQ8wc&usg=AFQjCNHsIqk4Z6puR4MNs44agEguLJJh_A

The Purpose of Flipping Inequality Symbols when Multiplying by a Negative Number and Solving for Absolute Value Inequalities!

Many of you solved absolute value inequality problems and regular inequality problems, but never understood why the inequality symbols must be flipped. Well...

Absolute Value Inequalities

The definition of absolute value is, the distance a number is from zero. Therefore, a negative number and a positive number have the same distance from zero. For example:

As you notice, -4 and 4 has the same absolute value. When solving for absolute value inequalities the sign is flipped because of the number line. On the number line both negative and positive numbers count because they both have a distance to Zero. 

Regular Inequality Problems:

Inequality problems on a number line show what values the variable is greater or less than. Therefore, when multiplying by a negative number the sign flips because it is actually less than what the variable can be. On a number line, the inequality sign must be flipped to satisfy the inequality. For Example,

As you can see, the flipping of the sign just makes the inequality true because on a number line a negative number is less than the variable or number used in the problem.

 

Question:

Solve for x: -3|x-9|>27. How many times did you flip the inequality sign?

Sources: 

• http://neaportal.k12.ar.us/wp-content/uploads/2010/08/sei2a14_solve-and-graph-absolute-value-equations-and-inequalities.jpg
• https://d3pof0dg6ipqi1.cloudfront.net/learning_preview/10913/image/large_hqdefault.jpg