How to Solve Quadratic Equations

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Allison Moffett, host of Mahalo's Math Channel, shows how to use the quadratic formula to solve quadratic equations. She explains the formula and provides a visual example by solving a sample quadratic equation. This is part of a series of instructional videos available on Mahalo.com's channel on YouTube.

If you are taking an intermediate
algebra class, you will have to know how to solve quadratic equations. This equation is in the form ax² + bx + c = 0. Quadratic equations are also known as equations of the second degree because they contain a term that is squared.http://library.thinkquest.org/20991/a...
A quadratic equation has two solutions, also known as roots. The reason for this is because the equation can be factored as (x − a)(x − b), with "a" and "b" being the two roots.
http://www.themathpage.com/alg/quadra... are two ways to solve a quadratic equation. One way is to factor the equation. The second is to use the quadratic formula. Both methods will produce the roots of the equation, but the quadratic formula is used most often in instances when the equation cannot be easily factored.http://www.purplemath.com/modules/qua...

Step 1: Put Equation in Standard Quadratic Form
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There will be equations that may not look like they are quadratic equations because they are not in the exact form of ax² + bx + c = 0. To make sure you are working with a quadratic equation, it is a good idea to see if the equation can be simplified to this form.

If you have an equation that reads 3x² = 6, it may seem uncertain if this is actually a quadratic equation. You think it is because it looks like a second degree equation do to the "x²" term. Try putting it into standard form

3x² - 6 = 0

You do not have a single "x" term. That does not mean that it is not a quadratic equation because zero times anything equals zero. So to put this into standard form, you would have:

3x² + 0x - 6 = 0.  

So 3x²  = 6 is a quadratic equation and you can solve it as such.
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Step 2: Using the Quadratic Formula
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For any equation that is in the form ax² + bx + c = 0, it can be solved using the
quadratic formula.  This formula is written:
x = [-b ± SQRT(b² - 4ac)]/2a

The same values of a, b and c are used as what appear in the quadratic equation.

Part of the quadratic formula is called the
discriminant. This is the part of the formula that reads (b² - 4ac). There are three different types of solutions for the discriminant and each will let you know how many solutions the quadratic equation has, if any.
If you plug in the values of a, b and c into the discriminant:

1. If the discriminant's solution is greater than zero, the equation has two solutions.
2. If it is zero, there is only one value that will be the solution.
3. If it is less than zero, no solutions are defined.
http://www.algebra.com/algebra/homewo...

Step 3: Solve the Quadratic Equation
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For the equation 3x² = 6, you can use the quadratic equation to solve it.  Before putting it in standard form, you should recognize that the equation can be simplified even further by dividing each side by 3.

3x²/3 =6/3

x² = 2

Put the equation in the standard form: x² + 0x - 2 = 0

You can find out how many solutions it will have using a = 1, b = 0 and c = -2.  The formula for the discriminant is -b -4ac, or for this equation it is 0 - 4(1)(-2) = 8.  Since 8 is greater than zero, this quadratic equation will have two solutions.

Using the values for a, b and c into the quadratic formula and putting the solution for the discriminant into the parentheses behind the square root, the solutions for the equation are:

x = [0 ± SQRT(8)]/2(1)

x = ± SQRT(8)/2

The square root of 8 is approximately equal to 2.8, so the two solutions to the equation are +2.8/2 and -2.8/2.  These solutions simplify to +1.4 and -1.4  you can plug each value into the original equation to check your work.
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