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Solving Quadratic Equations with the Quadratic Formula

By Edited Nov 13, 2013 0 0

woman pulling hair out

We have all been there before, trying to solve a quadratic equation that seems to impossible. Let me guess, you've tried: simplifying the equation, factoring, completing the square, and maybe even graphing, but nothing is working. In comes the quadratic formula:

quadratic formula(46099)

Looks scary! Well it isn't all that bad, and once you memorize that formula you can solve any quadratic equation in the world! But where do all those variables come from?

quadratic polynomial

That above is a one-variable quadratic polynomial. Yes, that is a mouthful. Let's take a second and dissect the exact meaning beginning with “one-variable.” Simple, this means that there can only be one variable in the entire equation, named x (in the above equation). But aren't the a, b, and c variables? No, they are not, we call them the coefficients. In layman terms a coefficient is simply something that  multiples with the variable, generally a constant that must NOT involve a variable. Now let's take a look at the meaning of “quadratic polynomial.” This means that we do not raise any variable to a power greater than 2. Pretty simple so far, now let's connect the two ideas.

The quadratic formula has an a, b, and c just like the quadratic equation so all we have to do is substitute the numbers from the quadratic equation into the quadratic formula and solve. Remember to include negatives and zeros as well! You will see in the examples below what I am talking about.

Once you solve the quadratic formula it will give you 2 answers. They can either be real, or complex numbers. If they are both answers are equal then that means the graph of the equation is tangent to the x-axis at that point. If both answers are complex numbers, then that means the graph of the equation never crosses the x-axis. Finally, if both answers are real numbers, then the graph of the equation crosses the x-axis 2 times, once at the first solution and another time at the second solution. Now let's work some problems!

Example 1

first example equation

a equals one

b equals 2
c equals negative three

Notice how the a is equal to one even though there is no coefficient in front of the x squared. This is because multiplying by 1 produces no effect, but we must include it when solving the quadratic formula! And once again, take note of how we include the negative sign when determining c. Now let's substitute these numbers into the quadratic formula and find out what we get.

solution one
solution two

Aha, the solutions are 1 and -3, therefore the graph of the equation must cross the x-axis at those two points. Does it?

graph of example one

It does, great! Whenever you work your own problems, it is a good idea to confirm your results by graphing the equation. Now let's have you solve two yourself. Feel free to post questions in the comment section.

Example 2

example two
 hint: 
hint one

Example 3

example three

Remember to confirm your solutions by graphing! Using the quadratic formula is an invaluable skill to posses. It is useful across a lot of fields such as chemistry, finance, business, and even art.


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