How was the Quadratic Formula Derived?

Note:

Ever wonder how mathematicians came up with the quadratic formula? Wonder no more! This tutorial takes you through the process of deriving the quadratic formula, so even if you forget it on an exam, you'll be able to derive it and then use it to solve your quadratic equation! Check it out!

Keywords:

quadratic formula
derivation
proof
equation
completing the square

Background Tutorials

Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
- How Do You Find a Common Denominator and a Least Common Denominator?
  
  This tutorial gives you some practice finding a common denominator and the least common denominator of three fractions. There's only one least common denominator, but there are many common denominators. This tutorial gives you one. Can you find another?
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.
- What's the Reciprocal Rule of Division?
  
  When you divide fractions, the trick is to rewrite division as a multiplication. But the truth is that you can always rewrite division as a multiplication, and in this tutorial you'll see the rule that makes that possible!
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.
- What's a Coefficient?
  
  There is a bunch of vocabulary that you just need to know when it comes to algebra, and coefficient is one of the key words that you have to feel 100% comfortable with. Check out the tutorial and let us know if you want to learn more about coefficients!
Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
- What's the Subtraction Property of Equality?
  
  Solving equations can be tough, especially if you've forgotten or have trouble understanding the tools at your disposal. One of those tools is the subtraction property of equality, and it lets you subtract the same number from both sides of an equation. Watch the video to see it in action!
- What's the Addition Property of Equality?
  
  Solving equations can be tough, especially if you've forgotten or have trouble understanding the tools at your disposal. One of those tools is the addition property of equality, and it lets you add the same number to both sides of an equation. Watch the video to see it in action!
Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
- How Do You Subtract Fractions with the Same Denominator?
  
  Subtracting fractions with like denominators? Just subtract the numerators and put the result over the common denominator! To see this process in action, check out this tutorial!
Factor a quadratic expression to reveal the zeros of the function it defines.
- How Do You Use a Shortcut to Factor a Perfect Square Trinomial?
  
  Factoring a perfect square trinomial? Did you know there's a shortcut to factoring this special kind of trinomial? Check it out! It's pretty cool, and it may make this process a little faster!
Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)^2= q that has the same solutions. Derive the quadratic formula from this form.
- How Do You Solve a Quadratic Equation by Completing the Square?
  
  One of the many ways you can solve a quadratic equation is by completing the square. In this method, you want to turn one side of the equation into a perfect square trinomial. This tutorial takes you through the steps of solving a quadratic equation by completing the square. Check it out!
- What is a Quadratic Equation?
  
  You can't go through algebra without seeing quadratic equations. The graphs of quadratic equations are parabolas; they tend to look like a smile or a frown. There's also a bunch of ways to solve these equations! Watch this tutorial and get introduced to quadratic equations!
- What is the Quadratic Formula?
  
  If you're solving quadratic equations, knowing the quadratic formula is a MUST! This formula is normally used when no other methods for solving quadratics can be reasonably used. In this tutorial, learn about the qu