What is the Quadratic Formula? – Definition, Equation & Proof

The quadratic formula is a method that is used to find the roots of a quadratic equation. In this lesson, you will learn about the history of the quadratic formula, how to use it, and prove it. A quiz at the end will solidify what you’ve learned.

Quadratic Equations

A quadratic equation is a polynomial in which the first term is of the second degree. This means that it is squared. The standard form of the quadratic equation is:

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There are different ways to find the roots, or solutions, of a quadratic equation. There are two solutions to every quadratic equation, however, they may or may not be real numbers. One way to find the solutions to this type of equation is by factoring. This is not always the best way to solve a quadratic equation, especially if the solutions are not whole numbers.

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The Quadratic Formula

One simple way to solve a quadratic equation is by using the quadratic formula. With just some simple math, you will have the solution quickly, whether the solution is a whole number, an irrational number, or even an imaginary number. The quadratic formula is:

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History of the Quadratic Formula

Babylonian mathematicians used a simple version of this formula as far back as 2000 B.C. The equation most closely related to the form we know today was first written down by a Hindu mathematician named Brahmagupta. Other slightly different forms followed in India and Persia.

European mathematics gained resurgence during the 1500s, and in 1545, Girolamo Cardano compiled all that was previously known about the quadratic formula and also allowed for the possibility of imaginary solutions. Ren; Descartes, the Father of Modern Mathematics, published the quadratic formula as we know it today in his book La G;om;trie.

Solving the Quadratic Formula

If you look again at the quadratic formula, you will see that there is a symbol that looks like a plus sign over a minus sign.

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This plus/minus sign means that there are really two equations to be solved, one with addition and one with subtraction.

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It’s written with the plus/minus sign to save space. To solve a quadratic equation using the quadratic formula, all you need to do is plug the proper numbers into their respective places and solve the two equations.

The number in front of x squared is a, the number in front of the x is b and the number by itself with no variable is c. If there is no number in front of either x squared or x, then use the number 1. If there is no x term or no number without a variable, use 0.

Example

Let’s solve the equation y = x^2 + 2x – 3

For this problem,

a = 1
b = 2
c = -3

If you plug all of the numbers into the formula, you will get:

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Now let’s figure for the two equations indicated by the plus/minus sign.

  • The first will be -2 plus 4 over 2, or 2 over 2, which equals 1!
  • The second is -2 minus 4, or -6 over 2, which equals -3!

Now look at this example – the solutions are not as easy; however, they are much easier than trying to solve by factoring the quadratic:

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Okay, so now we have x equals 1 plus or minus the square root of 5, over 2. The two equations indicated by the plus/minus sign are

  • 1 plus the square root of 5, over 2
  • 1 minus the square root of 5, over 2

Proof of the Quadratic Formula

This proof will show you how the quadratic formula was developed. It is performed using the standard form of a quadratic equation and solving it by completing the square.

Start with the standard form of a quadratic equation:

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Divide both sides of the equation by a so you can complete the square.

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Subtract c / a from both sides:

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Complete the square:

  • the coefficient of the second term is b / a
  • divide this coefficient by 2 and square the result to get (b / 2a)^2. This is completing the square.

Add (b/2a)^2 to both sides:

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The left side of the equation is a perfect square, so you can factor it by using the coefficient of the first term (x) and the base of the last term (b / 2a).

Add these two and raise everything to the second power.

Square the right side to get (b^2) / (4a^2):

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Get the same denominator on the right side:

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Now, take the square root of each side:

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Simplify the left side:

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Rewrite the right side:

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Subtract b / 2a from both sides:

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Adding the numerator and keeping the same denominator, we get the quadratic formula:

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Lesson Summary

The quadratic formula is one way to solve a quadratic equation. It can be easier than factoring, especially when the solutions are not whole numbers. All that is needed to find the solution is to substitute the specific numbers from the equation into the formula and solve. Just remember to split the formula into its two parts at the plus/minus sign. When you finish, you should have two solutions.

Lesson at a Glance

A quadratic equation is a polynomial in which the first term is of the second degree. Using the quadratic formula, you can solve for the quadratic equation by substituting the specific numbers from the equation into the formula and solving. You will have two solutions.

Using the quadratic formula will provide two solutions.
quadratic formula

Learning Outcomes

Study and accomplish these goals as soon as you feel ready:

  • Define ‘quadratic equation’
  • Write the quadratic formula
  • Recount its history
  • Utilize the quadratic formula to solve quadratic equations
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