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Using the Discriminant in Quadratics

Answer 1

The discriminant as it comes in Quadratics is the section of the quadratic formula that is \[b^{2} - 4ac\]This piece of the Quadratic Formula is very useful once you start getting into higher levels of math and when Quadratics becomes much quicker and easier. In fact, it can actually be useful whenever you come along a quadratic.
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The Basics
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Before we start going into how the discriminant works, let's go over the standard form for a quadratic equation.

\[ax^{2} + bx + c = 0\]
This is the standard form of a quadratic equation. You have to understand how to use a, b, and c in the quadratic formula in order to use the discriminant, so let's review with an example.

\[x^{2} + 2x - 3 = 0\]
Looking at the standard form of a quadratic again, you should be able to tell what a, b, and c are.

a = 1
b = 2
c = -3

Now that we are clear on what the a, b, and c's are in a quadratic equation, let's start actually learning about the discriminant itself.
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The Discriminant
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The purpose of the discriminant is to have an idea of what method to use in order to solve the quadratic equation. (Remember, there are 3 general non-graphing methods to solving quadratics which are factoring, the quadratic formula, and completing the square.) Let's take a look at how the discriminant affects how you would solve the quadratic.

If the discriminant is a positive number as well as a perfect square, then you should probably just factor to solve. (In terms of graphing, the 2 solutions will be rational numbers.)

If the discriminant is a positive number but not a perfect square, then you can't factor and should use the quadratic formula or complete the square. (In terms of graphing, the 2 solutions will be irrational numbers.)

If the discriminant is 0, then you should probably factor the equation. (In terms of graphing, this would be a double root meaning that the vertex is on the x-axis.)

If the discriminant is negative, then you should either use the quadratic formula or complete the square to solve. (In terms of graphing, the graph will never touch the x-axis because the roots will be imaginary.)

(Note about complete the square: never complete the square unless the leading coefficient is a perfect square.)
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Why It works
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The discriminant works like it does because if you recall, it is actually under a square root symbol. The square root of a perfect square is rational and therefore, factorable. The square root of a positive number that is not a perfect square results in an irrational number. The square root of 0 is 0 and therefore there is a double root. The square root of a negative number is an imaginary number.

Answer 2

nice :)

Answer 3

lol. I can't believe I made that mistake :)

Answer 4

Where is the edit button?

Answer 5

There we go :D

Answer 6

Thank you

Answer 7

Great tutorial : here is the one which i also made for quadratic equation .
http://openstudy.com/study#/updates/4fe08815e4b06e92b86fa074

for the proof

Answer 8

Good One ,Keep it up :)

Answer 9

Thanks everybody! @mathslover I will check that out :)

Answer 10

awesome