Solve a x^2 + b x + c = 0 for x

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When would I use the Quadratic Formula?

You can use the Quadratic Formula any time you"re trying to lớn solve a quadratic equation — as long as that equation is in the form "(a quadratic expression) that is mix equal lớn zero".

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Often, the simplest way lớn solve "ax2 + bx + c = 0" for the value of x is to lớn factor the quadratic, phối each factor equal lớn zero, và then solve each factor. But sometimes the quadratic is too messy, or it doesn"t factor at all, or, heck, maybe you just don"t feel like factoring. While factoring is not always going to lớn be successful, the Quadratic Formula can always find the answers for you.


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Practice The Quadratic Formula


The Quadratic Formula uses the "a", "b", & "c" from "ax2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients" of the quadratic equation they"ve given you to lớn solve.

The Quadratic Formula is derived from the process of completing the square, & is formally stated as:


What is the Quadratic Formula?

The Quadratic Formula is a rule that says that, in any equation of the size ax2 + bx + c = 0, the solution x-values of the equation are given by:


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How vày I use the Quadratic Formula?

To use the Quadratic Formula, you must:

Arrange your equation into the form "(quadratic) = 0".

Arrange the terms in the (equation) in decreasing order (so squared term first, then the x-term, and finally the linear term).

Pull out the numerical parts of each of these terms, which are the "a", "b", and "c" of the Formula.

Plug these numbers into the formula.

Simplify khổng lồ get your answers.

Advisories: The "2a" in the denominator of the Formula is underneath everything above, not just the square root. & it"s a "2a" under there, not just a plain "2". Make sure that you are careful not lớn drop the square root or the "plus/minus" in the middle of your calculations, or I can guarantee that you will forget lớn "put them back in" on your test, và you"ll mess yourself up. Remember that "b2" means "the square of ALL of b, including its sign", so don"t leave b2 being negative, even if b is negative, because the square of a negative is a positive.

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In other words, don"t be sloppy and don"t try khổng lồ take shortcuts, because it will only hurt you in the long run. Trust me on this!

What is an example of using the Quadratic Formula?

Solve x2 + 3x − 4 = 0

This quadratic happens to factor, which I can use to confirm what I get from the Quadratic Formula. The Formula should give me the same answers.

x2 + 3x − 4 = (x + 4)(x − 1) = 0

...so I already know that the solutions are x = −4 and x = 1.

Now, what would my solution look lượt thích in the Quadratic Formula? Using a = 1, b = 3, và c = −4, my solution process looks like this:


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So, as expected, the solution is x = −4, x = 1.


How does the Quadratic Formula relate khổng lồ x-intercepts?

The solutions lớn the quadratic equation, as provided by the Quadratic Formula, are the x-intercepts of the corresponding graphed parabola.

How? Well, when y = 0, you"re on the x-axis. The x-intercepts of the graph are where the parabola crosses the x-axis. You"re applying the Quadratic Formula lớn the equation ax2 + bx + c = y, where y is mix equal khổng lồ zero.

Looking at the above example, there were two solutions for the equation x2 + 3x − 4 = 0. This tells us that there must then be two x-intercepts on the graph. Graphing, we get the curve below:

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As you can see, the x-intercepts (the red dots above) match the solutions, crossing the x-axis atx = −4 & x = 1. This shows the connection between graphing and solving: When you are solving "(quadratic) = 0", you are finding the x-intercepts of the graph. This can be useful if you have a graphing calculator, because you can use the Quadratic Formula (when necessary) lớn solve a quadratic, and then use your graphing calculator to make sure that the displayed x-intercepts have the same decimal values as bởi the solutions provided by the Quadratic Formula.

Note, however, that the calculator"s display of the graph will probably have some pixel-related round-off error, so you"d be checking to see that the computed & graphed values were reasonably close; don"t expect an exact match.

Solve 2x2 − 4x − 3 = 0. Round your answer to lớn two decimal places, if necessary.

There are no factors of (2)(−3) = −6 that địa chỉ up lớn −4, so I know that this quadratic cannot be factored. I will apply the Quadratic Formula. In this case, a = 2, b = −4, & c = −3:


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Can I round my answers from the Quadratic Formula?

In general, no, you really shouldn"t; the "solution" or "roots" or "zeroes" of a quadratic are usually required khổng lồ be in the "exact" khung of the answer. You can use the rounded size when graphing (if necessary), but "the answer(s)" from the Quadratic Formula should be written out in the (often messy) "exact" form.

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In the example above, the exact size is the one with the square roots of ten in it. If you"re wanting to graph the x-intercepts or needing lớn simplify the final answer in a word problem to lớn be of a practical ("real world") form, then you can use the calculator"s approximation. But unless you have a good reason lớn think that the answer is supposed to be a rounded answer, always go with the exact form.

Reinforcing the concept: Compare the solutions we found above for the equation 2x2 − 4x − 3 = 0 with the x-intercepts of the graph:

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Just as in the previous example, the x-intercepts match the zeroes from the Quadratic Formula. This is always true. The "solutions" of an equation are also the x-intercepts of the corresponding graph.
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