Content

• To step
• Method 1 of 2: Part One: Rewriting a standard equation
• Method 2 of 2: Part Two: Solving a Quadratic Equation
• Tips

Squaring off is a useful technique for writing a quadratic equation differently, making it easier to survey and solve. You can rewrite a square by rearranging it into more manageable pieces.

To step

Method 1 of 2: Part One: Rewriting a standard equation

1. Write down the equation. Let's say you want to solve the following equation: 3x - 4x + 5.
2. Get the coefficient from the equation. Place the 3 outside parentheses and divide each term, except the constant, by 3. 3x divided by 3 is x and 4x divided by 3 is 4 / 3x. So the new equation looks like this: 3 (x - 4 / 3x) + 5. The 5 is outside the parentheses because you didn't divide it by 3.
3. Divide the second term by 2 and square. The second term, also called the bterm in the equation is 4/3. Halve the second term. 4/3 ÷ 2, or 4/3 x 1/2, equals 2/3. Square this term by multiplying both the numerator and denominator by themselves. (2/3) = 4/9. Write this term down.
4. Addition and subtraction. You need this "extra" term to convert the first three terms of the equation into a square. But keep in mind that you added this term by subtracting it from the equation as well. Of course, it makes little difference to simply put the terms back together - then you go back to where you started. The new equation should now look like this: 3 (x - 4/3 x + 4/9 - 4/9) + 5.
5. Take the term you subtracted outside of parentheses. Because you are already working with the 3 outside the brackets, it is not possible to just put -4/9 outside the brackets. First you have to multiply it by 3. -4/9 x 3 = -12/9, or -4/3. If you are dealing with an equation that only contains a coefficient 1 of x, you can skip this step.
6. Convert the terms in parentheses to a square. Your equation now looks like this: 3 (x -4 / 3x +4/9). You worked front to back to get 4/9, which is actually another way to find the factor that completes the square. So you can rewrite these terms as: 3 (x - 2/3). You can check this by multiplying and you will see that you get the same original equation as the answer again.
   • 3 (x - 2/3) =
   • 3 (x - 2/3) (x -2/3) =
   • 3 [(x -2 / 3x -2 / 3x + 4/9)]
   • 3 (x - 4 / 3x + 4/9)
7. Merge the constants. You now have two constants, 3 (x - 2/3) - 4/3 + 5. All you have to do now is add -4/3 to 5 and this will give you 11/3 as the answer. You do this by giving them the same denominator: -4/3 and 15/3, and then adding both numerators to get 11, leaving the denominator equal to 3.
   • -4/3 + 15/3 = 11/3.
8. Write the equation in a different form. Now you are done. The final equation is 3 (x - 2/3) + 11/3. You can eliminate the 3 by dividing the equation by 3, after which you are left with the following equation: (x - 2/3) + 11/9. You have now successfully written the equation in a different form: a (x - h) + k, at which k is the constant.

Method 2 of 2: Part Two: Solving a Quadratic Equation

1. Write down the statement. Let's say you want to solve the following equation: 3x + 4x + 5 = 6
2. Add the constants and place them to the left of the equal sign. Constant terms are those terms without a variable. In this case, you have 5 on the left and 6 on the right. You want to move 6 to the left, so subtract 6 from both sides of the equation. That leaves 0 on the right (6-6) and -1 on the left (5-6). The equation now looks like this: 3x + 4x - 1 = 0.
3. Exclude the coefficient of the square from parentheses. In this case, 3 is the coefficient of x. To put 3 out of parentheses, remove 3, put the remaining term in parentheses, and divide each term by 3. So, 3x ÷ 3 = x, 4x ÷ 3 = 4 / 3x, and 1 ÷ 3 = 1/3. The equation now looks like this: 3 (x + 4 / 3x - 1/3) = 0.
4. Divide by the constant you just put out of parentheses. This will finally get you rid of those pesky 3 outside the brackets. Because you divide each term by 3, it can be eliminated without changing the equation. Now you have: x + 4 / 3x - 1/3 = 0
5. Divide the second term by 2 and square. Take the second term, 4/3, the b term, and divide by 2. 4/3 ÷ 2 or 4/3 x 1/2, is 4/6, or 2/3. And 2/3 squared is 4/9. When you are done with this, you should write it to the left and right of the equation because you really just added a new term. You have to do this on both sides of the equation. The equation now looks like this: x + 4/3 x + 2/3 - 1/3 = 2/3
6. Move the original constant to the right side of the equation and add it to the term that is already there. Move the constant, -1/3, to the right to make it 1/3. Add these to the other term, 4/9, or 2/3. Find the least common multiple so that 1/3 and 4/9 can be added together. This is done as follows: 1/3 x 3/3 = 3/9. Now add 3/9 to 4/9 so that you have 7/9 to the right of the equation. This gives: x + 4/3 x + 2/3 = 4/9 + 1/3 and then x + 4/3 x + 2/3 = 7/9.
7. Write the left side of the equation as a square. Since you've already used a formula to find the missing term, the trickiest part has already been done. All you have to do is put the x and half of the second coefficient in parentheses and square it, like this: (x + 2/3). Note that factoring the square yields 3 terms: x + 4/3 x + 4/9. The equation now looks like this: (x + 2/3) = 7/9.
8. Take the square root of both sides of the equation. On the left side of the equation, the square root of (x + 2/3) equals x + 2/3. The right side gives +/- (√7) / 3. The square root of the denominator 9 is 3, and the square root of 7 is √7. Don't forget to write the +/- because a square root of a number can be positive or negative.
9. Set the variable aside. To isolate the variable x from the rest, move the constant 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 this as is or elaborate on the square root, if you are asked for an answer without a square root sign.

Tips

• Make sure you put the +/- in the right places otherwise you will only get one answer.
• Even if you know the square root formula, it doesn't hurt to practice splitting off the squares or working out quadratic equations from time to time. That way you can be sure that you know how to do it when necessary.