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Completing the Square Provided by the Academic Center for Excellence 1 Reviewed August, 2014
Completing the Square
โCompleting the squareโ is another method of solving quadratic equations. It allows trinomials to be factored into two identical factors.
Example: ๐ฅ๐ฅ^2 + 4๐ฅ๐ฅ + 4 (๐ฅ๐ฅ + 2)(๐ฅ๐ฅ + 2) or (๐ฅ๐ฅ + 2)^2
To complete the square, it is necessary to find the constant term, or the last number that will enable factoring of the trinomial into two identical factors. To find the constant term needed, simply take the coefficient of โ๐ฅ๐ฅ,โ divide by 2 , and square the quotient. If you have an equation, rather than an expression, the resulting number should be added to both sides of the equation.
Example: What is the constant term used to factor the expression ๐ฅ๐ฅ^2 โ 8 ๐ฅ๐ฅ into two identical factors? Step 1. Take the coefficient of โ๐ฅ๐ฅโ, which is โ 8 , and divide it by two. โ 2 =^ โ^4 Step 2. Take that number and square it. (โ4)^2 = 16 Step 3. Adding the constant term of 16 would allow the expression to be factored into identical factors. ๐ฅ๐ฅ^2 โ 8 ๐ฅ๐ฅ + 16 = (๐ฅ๐ฅ โ 4)^2
To solve an equation by completing the square requires a couple of extra steps.
Example: Solve by completing the square ๐ฅ๐ฅ^2 + 8๐ฅ๐ฅ + 7 = 0 Step 1. Move the constant term to the other side of the equation by subtracting from both sides. ๐ฅ๐ฅ^2 + 8๐ฅ๐ฅ + 7 โ 7 = 0 โ 7 ๐ฅ๐ฅ^2 + 8๐ฅ๐ฅ = โ 7
Step 2. Complete the square.
= 4^2 = 16
Step 3. Since 16 is being added to the left side of the equation it MUST also be added to the right side. ๐ฅ๐ฅ^2 + 8๐ฅ๐ฅ + 16 = โ7 + 16 ๐ฅ๐ฅ^2 + 8๐ฅ๐ฅ + 16 = 9 Step 4. Factor the left side of the equation. ๐ฅ๐ฅ^2 + 8๐ฅ๐ฅ + 16 = 9 (๐ฅ๐ฅ + 4)^2 = 9 HINT: the number inside the factor should always be the same as the number obtained from dividing the coefficient of โxโ by two! 8 2 = 4^ and the factor was^ (๐ฅ๐ฅ^ + 4)
2
Step 5. Take the square root of both sides and solve for ๐ฅ๐ฅ.
๏ฟฝ^2 (๐ฅ๐ฅ + 4)^2 = (^) โ^29 ๐ฅ๐ฅ + 4 = ยฑ ๐ฅ๐ฅ = โ7 and โ 1
To complete the square, the coefficient of ๐ฅ๐ฅ^2 must be one. If it is any other number, first divide the entire equation by that number.
Example: Solve by completing the square 4 ๐ฅ๐ฅ^2 โ 12 ๐ฅ๐ฅ โ 4 = 12 Step 1. Divide the equation by 4 in order to get a leading coefficient of 1. (4x^2 โ 12x โ 4) 4 =
12 4 ๐ฅ๐ฅ^2 โ 3 ๐ฅ๐ฅ โ 1 = 3
Practice Problems
- ๐ฅ๐ฅ = โ1 and ๐ฅ๐ฅ =
- ๐ฅ๐ฅ^2 + 6๐ฅ๐ฅ + 5 =
- ๐ฅ๐ฅ^2 + 8๐ฅ๐ฅ โ 9 =
- ๐ฅ๐ฅ^2 โ 6๐ฅ๐ฅ + 9 =
- ๐ฅ๐ฅ 2 + 4๐ฅ๐ฅ โ 7 =
- ๐ฅ๐ฅ 2 โ 5๐ฅ๐ฅ โ 24 =
- ๐ฅ๐ฅ^2 โ 8๐ฅ๐ฅ + 15 =
- 4 ๐ฅ๐ฅ 2 โ 4๐ฅ๐ฅ + 17 =
- 9 ๐ฅ๐ฅ^2 โ 12๐ฅ๐ฅ + 13 =
- 4 ๐ฅ๐ฅ^2 โ 4๐ฅ๐ฅ + 5 =
- 4 ๐ฅ๐ฅ^2 โ 8๐ฅ๐ฅ + 1 =
Answers to Practice Problems
โ5 and โ 1
1 and โ 9
3 only
โ 2 โ โ 11 and โ 2 + โ 11
โ3 and 8
3 and 5
2 + 2๐๐^ and^
2 โ^2 ๐๐
3 +^ ๐๐^ and^
2 +^ ๐๐^ and^
2 and 1^ โ^
