One of the most important formulas for an algebra student is the quadratic one, that is x = (- b ± √ (b2 - 4ac)) / 2a. With this formula, to solve quadratic equations (equations in the form x2 + bx + c = 0) just substitute the values of a, b and c. While knowing the formula is often enough for most people, understanding how it was derived is another matter. In fact, the formula is derived with a useful technique called "square completion" which has other mathematical applications as well.
Steps
Method 1 of 2: Derive the Formula
Step 1. Start with a quadratic equation
All quadratic equations have the form ax2 + bx + c = 0. To start deriving the quadratic formula, simply write this general equation on a sheet of paper, leaving plenty of space under it. Don't substitute any numbers for a, b, or c - you'll work with the general form of the equation.
The word "quadratic" refers to the fact that the term x is squared. Whatever the coefficients used for a, b, and c, if you can write an equation in the normal binomial form, it is a quadratic equation. The only exception to this rule is "a" = 0 - in this case, since the term x is no longer present2, the equation is no longer quadratic.
Step 2. Divide both sides by "a"
To get the quadratic formula, the goal is to isolate "x" on one side of the equal sign. To do this, we will use the basic "erasing" techniques of algebra, to gradually move the rest of the variables to the other side of the equal sign. Let's start by simply dividing the left side of the equation by our variable "a". Write this under the first line.
- When dividing both sides by "a," don't forget the distributive property of divisions, which means that dividing the entire left side of the equation by a is like dividing terms individually.
- This gives us x2 + (b / a) x + c / a = 0. Note that the a multiplying the term x2 has been cleared and that the right side of the equation is still zero (zero divided by any number other than zero equals zero).
Step 3. Subtract c / a from both sides
As a next step, delete the non-x term (c / a) from the left side of the equation. Doing this is easy - just subtract it from both sides.
In doing so it remains x2 + (b / a) x = -c / a. We still have the two terms in x on the left, but the right side of the equation is starting to take the desired shape.
Step 4. Sum b2/ 4a2 from both sides.
Here things get more complex. We have two different terms in x - one squared and one simple - on the left side of the equation. At first glance, it might seem impossible to keep simplifying because the rules of algebra prevent us from adding variable terms with different exponents. A "shortcut", however, called "completing the square" (which we will discuss shortly) allows us to solve the problem.
- To complete the square, add b2/ 4a2 on both sides. Remember that the basic rules of algebra allow us to add almost anything on one side of the equation as long as we add the same element on the other, so this is a perfectly valid operation. Your equation should now look like this: x2+ (b / a) x + b2/ 4a2 = -c / a + b2/ 4a2.
- For a more detailed discussion of how square completion works, read the section below.
Step 5. Factor the left side of the equation
As a next step, to handle the complexity we just added, let's just focus on the left side of the equation for one step. The left side should look like this: x2+ (b / a) x + b2/ 4a2. If we think of "(b / a)" and "b2/ 4a2"as a simple coefficients" d "and" e ", respectively, our equation has, in effect, the form x2 + dx + e, and can therefore be factored into (x + f)2, where f is 1/2 of d and the square root of e.
- For our purposes, this means that we can factor the left side of the equation, x2+ (b / a) x + b2/ 4a2, in (x + (b / 2a))2.
- We know this step is correct because (x + (b / 2a))2 = x2 + 2 (b / 2a) x + (b / 2a)2 = x2+ (b / a) x + b2/ 4a2, the original equation.
- Factoring is a valuable algebra technique that can be very complex. For a more in-depth explanation of what factoring is and how to apply this technique, you can do some research on the internet or wikiHow.
Step 6. Use the common denominator 4a2 for the right side of the equation.
Let's take a short break from the complicated left side of the equation and find a common denominator for the terms on the right. To simplify the fractional terms on the right, we need to find this denominator.
- This is quite easy - just multiply -c / a by 4a / 4a to get -4ac / 4a2. Now, the terms on the right should be - 4ac / 4a2 + b2/ 4a2.
- Note that these terms share the same denominator 4a2, so we can add them to get (b2 - 4ac) / 4a2.
- Remember that we don't have to repeat this multiplication on the other side of the equation. Since multiplying by 4a / 4a is like multiplying by 1 (any non-zero number divided by itself equals 1), we are not changing the value of the equation, so there is no need to compensate from the left side.
Step 7. Find the square root of each side
The worst is over! Your equation should now look like this: (x + b / 2a)2) = (b2 - 4ac) / 4a2). Since we are trying to isolate x from one side of the equal sign, our next task is to calculate the square root of both sides.
In doing so it remains x + b / 2a = ± √ (b2 - 4ac) / 2a. Don't forget the ± sign - negative numbers can also be squared.
Step 8. Subtract b / 2a from both sides to finish
At this point, x is almost alone! Now, all that's left to do is subtract the term b / 2a from both sides to isolate it completely. Once finished, you should get x = (-b ± √ (b2 - 4ac)) / 2a. Does it look familiar to you? Congratulations! You got the quadratic formula!
Let's analyze this last step further. Subtracting b / 2a from both sides gives us x = ± √ (b2 - 4ac) / 2a - b / 2a. Since both b / 2a let √ (b2 - 4ac) / 2a have as common denominator 2a, we can add them, obtaining ± √ (b2 - 4ac) - b / 2a or, with easier reading terms, (-b ± √ (b2 - 4ac)) / 2a.
Method 2 of 2: Learn the "Completing the Square" Technique
Step 1. Start with the equation (x + 3)2 = 1.
If you didn't know how to derive the quadratic formula before you started reading, you are probably still a little confused by the "completing the square" steps in the previous proof. Don't worry - in this section, we'll break down the operation in more detail. Let's start with a fully factored polynomial equation: (x + 3)2 = 1. In the following steps, we will use this simple example equation to understand why we need to use "square completion" to get the quadratic formula.
Step 2. Solve for x
Solve (x + 3)2 = 1 times x is pretty simple - take the square root of both sides, then subtract three from both to isolate x. Read below for a step-by-step explanation:
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(x + 3)2 = 1
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- (x + 3) = √1
- x + 3 = ± 1
- x = ± 1 - 3
- x = - 2, -4
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Step 3. Expand the equation
We solved for x, but we're not done yet. Now, let's "open" the equation (x + 3)2 = 1 writing in long form, like this: (x + 3) (x + 3) = 1. Let's expand this equation again, multiplying the terms in parentheses together. From the distributive property of multiplication, we know we have to multiply in this order: the first terms, then the external terms, then the internal terms, finally the last terms.
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Multiplication has this development:
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- (x + 3) (x + 3)
- (x × x) + (x × 3) + (3 × x) + (3 × 3)
- x2 + 3x + 3x + 9
- x2 + 6x + 9
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Step 4. Transform the equation into quadratic form
Now our equation looks like this: x2 + 6x + 9 = 1. Note that it is very similar to a quadratic equation. To get the complete quadratic form, we just need to subtract one from both sides. So we get x2 + 6x + 8 = 0.
Step 5. Let's recap
Let's review what we already know:
- The equation (x + 3)2 = 1 has two solutions for x: -2 and -4.
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(x + 3)2 = 1 is equal to x2 + 6x + 9 = 1, which is equal to x2 + 6x + 8 = 0 (a quadratic equation).
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- Therefore, the quadratic equation x2 + 6x + 8 = 0 has -2 and -4 as solutions for x. If we verify by substituting these solutions for x, we always get the correct result (0), so we know that these are the right solutions.
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Step 6. Learn the general techniques of "completing the square"
As we saw earlier, it is easy to solve quadratic equations by taking them into the form (x + a)2 = b. However, to be able to bring a quadratic equation into this convenient form, we may have to subtract or add a number on both sides of the equation. In the most general cases, for quadratic equations in the form x2 + bx + c = 0, c must be equal to (b / 2)2 so that the equation can be factored into (x + (b / 2))2. If not, just add and subtract numbers on both sides to get this result. This technique is called "square completion", and that's exactly what we did to get the quadratic formula.
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Here are other examples of quadratic equation factorizations - note that, in each, the term "c" equals the term "b" divided by two, squared.
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- x2 + 10x + 25 = 0 = (x + 5)2
- x2 - 18x + 81 = 0 = (x + -9)2
- x2 + 7x + 12.25 = 0 = (x + 3.5)2
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Here is an example of a quadratic equation where the term "c" is not equal to half of the term "b" squared. In this case, we would have to add to each side to get the desired equality - in other words, we need to "complete the square".
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- x2 + 12x + 29 = 0
- x2 + 12x + 29 + 7 = 0 + 7
- x2 + 12x + 36 = 7
- (x + 6)2 = 7
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