Completing the square is a useful technique that allows you to reorganize an equation in a form that is easy to visualize or even to solve. You can complete the square to avoid using a complicated formula or to solve a second degree equation. If you want to know how, just follow these steps.
Steps
Method 1 of 2: Transforming an Equation from Standard Shape to Parabolic Shape with Vertex
Step 1. Consider the 3 x problem as an example2 - 4 x + 5.
Step 2. Collect the squared term coefficient from the first two monomials
In the example we collect a three and, putting a parenthesis, we get: 3 (x2 - 4/3 x) + 5. The 5 stays out because you don't divide it by 3.
Step 3. Halve the second term and square it
The second term, also known as term b of the equation, is 4/3. Halve it. 4/3 ÷ 2 or 4/3 x ½ is equal to 2/3. Now square the numerator and denominator of this fractional term. (2/3)2 = 4/9. Write it down.
Step 4. Add and subtract this term
Remember that adding 0 to an expression does not change its value, so you can add and subtract the same monomial without affecting the expression. Add and subtract 4/9 inside the parenthesis to get the new equation: 3 (x2 - 4/3 x + 4/9 - 4/9) + 5.
Step 5. Take the term you subtracted out of the parenthesis
You won't take out -4/9, but you will multiply it by 3. -4/9 x 3 = -12/9 or -4/3 first. If the coefficient of the second degree term x2 is 1, skip this step.
Step 6. Convert the terms in parentheses to a perfect square
Now you end up with 3 (x2 -4 / 3x +4/9) in parentheses. You found 4/9, which is another way to find the term that completes the square. You can rewrite these terms like this: 3 (x - 2/3)2. You have halved the second term and removed the third. You can do the test by multiplying, to check if you find all the terms of the equation.
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3 (x - 2/3)2 =
Complete the Square Step 6Bullet1 - 3 (x - 2/3) (x -2/3) =
- 3 [(x2 -2 / 3x -2 / 3x + 4/9)]
- 3 (x2 - 4 / 3x + 4/9)
Step 7. Put the constant terms together
You have 3 (x - 2/3)2 - 4/3 + 5. You have to add -4/3 and 5 to get 11/3. In fact, bringing the terms to the same denominator 3, we get -4/3 and 15/3, which together make 11/3.
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-4/3 + 15/3 = 11/3.
Complete the Square Step 7Bullet1
Step 8. This gives rise to the quadratic form of the vertex, which is 3 (x - 2/3)2 + 11/3.
You can remove the coefficient 3 by dividing both parts of the equation, (x - 2/3)2 + 11/9. You now have the quadratic form of the vertex, which is a (x - h)2 + k, where k represents the constant term.
Method 2 of 2: Solving a Quadratic Equation
Step 1. Consider the 3x second degree equation2 + 4x + 5 = 6
Step 2. Combine the constant terms and put them on the left side of the equation
Constant terms are all those terms that are not associated with a variable. In this case, you have 5 on the left side and 6 on the right side. You have to move 6 to the left, so you have to subtract it from both sides of the equation. This way you will have 0 on the right side (6 - 6) and -1 on the left side (5 - 6). The equation should now be: 3x2 + 4x - 1 = 0.
Step 3. Collect the coefficient of the squared term
In this case it is 3. To collect it, just extract a 3 and put the remaining terms in brackets dividing them by 3. So you have: 3x2 ÷ 3 = x2, 4x ÷ 3 = 4 / 3x and 1 ÷ 3 = 1/3. The equation has become: 3 (x2 + 4 / 3x - 1/3) = 0.
Step 4. Divide by the constant you just collected
This means you can permanently get rid of that 3 out of the bracket. Since each member of the equation is divided by 3, it can be removed without compromising the result. We now have x2 + 4 / 3x - 1/3 = 0
Step 5. Halve the second term and square it
Next, take the second term, 4/3, known as the b term, and divide it in half. 4/3 ÷ 2 or 4/3 x ½ is 4/6 or 2/3. And 2/3 squared gives 4/9. When you are done, you will have to write it on the left And to the right of the equation, since you are essentially adding a new term and, to keep the equation balanced, it must be added to both sides. We now have x2 + 4/3 x + (2/3)2 - 1/3 = (2/3)2
Step 6. Move the constant term to the right side of the equation
To the right it will do + 1/3. Add it to 4/9, finding the lowest common denominator. 1/3 will become 3/9 you can add it to 4/9. Added together they give 7/9 on the right side of the equation. At this point we will have: x2 + 4/3 x + 2/32 = 4/9 + 1/3 and therefore x2 + 4/3 x + 2/32 = 7/9.
Step 7. Write the left side of the equation as a perfect square
Since you have already used a formula to find the missing term, the already passed the hardest part. All you have to do is insert the x and half of the second coefficient in brackets, squaring them. We will have (x + 2/3)2. Squaring we will get three terms: x2 + 4/3 x + 4/9. The equation, now, should be read as: (x + 2/3)2 = 7/9.
Step 8. Take the square root of both sides
On the left side of the equation, the square root of (x + 2/3)2 it is simply x + 2/3. On the right, you'll get +/- (√7) / 3. The square root of the denominator, 9, is simply 3 and of 7 is √7. Remember to write +/- because the square root of a number can be positive or negative.
Step 9. Isolate the variable
To isolate the variable x, move the constant term 2/3 to the right side of the equation. You now have two possible answers for x: +/- (√7) / 3 - 2/3. These are your two answers. You can leave them like this or calculate the approximate square root of 7 if you have to give an answer without the radical sign.
Advice
- Make sure you put the + / - in the appropriate place, otherwise you will only get a solution.
- Even if you know the formula, periodically practice completing the square, proving the quadratic formula, or solving some practical problems. This way you won't forget how to do it when you need it.
