Math Solver

Exact vertex form

Complete the Square Calculator

Turn a rational quadratic into a scaled square plus a constant, or solve an equation through the same method. Every displayed identity and root set is checked independently.

Complete a squareSee the method
Status
Verified
Maximum degree
2
Domain
Real numbers
Independent check
Identity + root set

One method, two useful outcomes

For an expression, the result is exact vertex form. For an equation, the squared binomial is isolated and both real square-root branches are checked against the original equation.

Unit leading coefficient

x2+6x+5=(x+3)2−4x^2+6x+5=(x+3)^2-4

Non-unit leading coefficient

2x2+8x+3=2(x+2)2−52x^2+8x+3=2(x+2)^2-5

Solve through the square

x2+6x+5=0⇒(x+3)2=4x^2+6x+5=0\Rightarrow(x+3)^2=4

Current limitations

  • Select Complete square explicitly in the solver method control.
  • One variable and exact rational coefficients only.
  • No inequalities, variable denominators, or degree above two.
  • Only real equation roots are returned.

Questions

Complete the square FAQ

What form does the calculator return?

A quadratic expression is returned as a(x-h)^2+k with exact rational values. A quadratic equation shows the corresponding squared-binomial equation before returning all real roots.

Does it work when the leading coefficient is not one?

Yes. The leading coefficient may be any nonzero exact rational value within the solver resource limits.

Does it support inequalities or higher-degree polynomials?

No. This mode is limited to one-variable quadratic expressions and equations. Unsupported forms are rejected rather than approximated.