Free · Algebra and geometric models

Binomial Squares and Both Real Roots

Expand sum or difference squares and inspect both solutions of a squared linear expression.

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

The square (2x+3)² expands to 4x²+12x+9 and touches the horizontal axis at x=−1.5.

(a±b)²=a²±2ab+b²; (fx+g)²=y gives x=(−g±√y)/f

Worked example

Enter these known values and leave the other values blank.

Choose the calculation that you would like to perform:
2
Insert your *a*
2
Insert your *b*
3
Result
(a±b)²=a²±2ab+b²; (fx+g)²=y gives x=(−g±√y)/f.
Explanation
(a±b)²=a²±2ab+b²; (fx+g)²=y gives x=(−g±√y)/f.
Explanation
(a±b)²=a²±2ab+b²; (fx+g)²=y gives x=(−g±√y)/f. Insert your *a* = 2. Insert your *b* = 3.
Explanation
(a±b)²=a²±2ab+b²; (fx+g)²=y gives x=(−g±√y)/f. Insert your *a* = 2. Insert your *a* = 2. Insert your *b* = 3. Insert your *b* = 3.
Explanation
(a±b)²=a²±2ab+b²; (fx+g)²=y gives x=(−g±√y)/f. F sq = 4. Fg times 2 = 12. G sq = 9.
Explanation
(a±b)²=a²±2ab+b²; (fx+g)²=y gives x=(−g±√y)/f. Xintercept = -1.5. G sq = 9.

Assumptions and limitations

  • The real target y of a squared expression is nonnegative.
  • Both roots are shown when an inverse admits two solutions. The linear coefficient f must be nonzero to solve for x.

Common questions

What must I enter?

Choose the mode when available and enter its independent known quantities. Leave an unknown editable quantity empty to use a supported inverse.

How should I use the result?

Interpret the numbers within the stated mathematical assumptions. Available unit selectors convert quantities; display rounding does not replace the internal calculation.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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