What Is Add the Proper Constant to the Binomial?
To add the proper constant to a binomial such as x² + 8x, take half of the x coefficient and square it. Adding that constant turns the binomial into a perfect square trinomial: x² + 8x + 16 = (x + 4)².
Enter the coefficient of x² (use 1 when there is none) and the coefficient of x. The calculator shows the constant, the full trinomial and its factored square, using exact fractions when the coefficient is odd.
When the leading coefficient is not 1, factor it out of the x terms first. The constant you add to the whole expression is then the leading coefficient times the square of half the new coefficient.
How to Use the Add the Proper Constant to the Binomial Calculator
- Enter the coefficient of x² (use 1 if there is none) and the coefficient of x.
- Read the constant to add.
- Use the factored square in the final line.
Add the Proper Constant to the Binomial Formula
Constant for a = 1
General a
Add the Proper Constant to the Binomial Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| a | Coefficient of x² | |
| b | Coefficient of x | |
| b/2 | Half the x coefficient, which sits inside the squared binomial | |
| (b/2)² | The constant that completes the square | Always positive |
Add the Proper Constant to the Binomial Worked Example
With the example values loaded in the calculator above, the steps are:
- Start with x² + 8x.
- Take half of the x coefficient: 8 ÷ 2 = 4.
- Square it: (4)² = 16.
- Add 16 to get x² + 8x + 16, which factors as shown.
Result: Constant to add = 16; Perfect square trinomial = x² + 8x + 16.
Worked example: x² + 8x
Half of 8 is 4, and 4² = 16. Adding 16 gives x² + 8x + 16 = (x + 4)².
Check: (x + 4)² = x² + 8x + 16.
When the leading coefficient is not 1
For 2x² + 12x, factor out 2 first: 2(x² + 6x). Half of 6 is 3 and 3² = 9, so inside the bracket you add 9. Because of the factor 2, the constant added to the whole expression is 2 × 9 = 18, giving 2x² + 12x + 18 = 2(x + 3)².
Where Add the Proper Constant to the Binomial Is Used
Completing the square
Do the key step when solving quadratics or finding the vertex.
Graphing parabolas
Rewrite a quadratic in vertex form.
Integration and conic sections
Complete the square before integrating or finding a circle's centre.
Common Add the Proper Constant to the Binomial Mistakes to Avoid
- In an equation, add the constant to both sides.
- A negative x coefficient gives a minus sign in the square, but the constant is still positive.
Add the Proper Constant to the Binomial Terms and Definitions
- Perfect square trinomial
- A trinomial that is the square of a binomial.
Add the Proper Constant to the Binomial FAQs
How do you find the proper constant to add to a binomial?
Halve the coefficient of x and square it.
What constant should be added to x² + 6x?
9, because (6 ÷ 2)² = 9, and x² + 6x + 9 = (x + 3)².
Does the sign of b change the constant?
No. The constant is a square, so it is always positive. The sign shows up inside the squared binomial.
What if the leading coefficient is not 1?
Factor it out of the x terms, complete the square inside the bracket, then multiply the constant by the factor.
Why is it called completing the square?
The added constant completes a square whose side is (x + b/2).
Can the constant be a fraction?
Yes. If b is odd, the constant is a fraction such as 25/4. This calculator keeps exact fractions.