Completing the square turns ax² + bx + c into vertex form a(x − h)² + k.
About this calculator
Completing the square turns ax² + bx + c into vertex form a(x − h)² + k. The vertex sits at (h, k). For x² − 6x + 5, h = 3 and k = −4, so the parabola bottoms out at (3, −4).
The same algebra behind vertex form also feeds the quadratic formula. Discriminant b² − 4ac tells you whether roots are real: positive gives two, zero gives one, negative gives none in the reals.
For full root-finding with complex numbers use [[quadratic]]. For linear cases when a = 0 try [[linear-equation]]. FOIL expansion checks go through [[foil-method]].
Common uses
- Convert x² − 6x + 5 to vertex form
- Find the vertex of a parabola for graphing
- Check discriminant before solving quadratics
- Prepare completing-the-square homework steps
- Locate min or max of a quadratic model
Frequently asked questions
What is the vertex of x² − 6x + 5?
h = −(−6)/(2×1) = 3. k = 5 − (−6)²/4 = 5 − 9 = −4. Vertex: (3, −4).
What does a negative discriminant mean?
b² − 4ac < 0 means no real roots. The parabola never crosses the x-axis.
Can a be zero?
No. a = 0 makes the equation linear, not quadratic. Use the linear equation calculator instead.
How is vertex form used in graphing?
Plot the vertex (h, k), then use a to sketch opening up or down. Symmetry about x = h makes plotting fast.