Completing The Square Works - Step by Step Algebra

Completing the square works is a guided walk-through of the algebraic technique that rewrites any quadratic ax² + bx + c = 0 as a perfect square trinomial plus a constant, and from there isolates the variable to produce the roots. The method is general: it works for monic and non-monic quadratics, for perfect square trinomials, and for quadratics whose discriminant is positive, zero, or negative. This page makes the underlying identity explicit so you can follow each step instead of trusting the formula on faith.

• • Prove the method always works: Test several quadratics and confirm the same five steps produce a complete answer every time.
• • Learn the algebra visually: Watch b / 2a appear as the half that turns the left side into a perfect square, then read the roots.
• • Derive the quadratic formula from scratch: Run the derivation with a, b, c as symbols and see x = (-b ± √(b² - 4ac)) / (2a) appear as a special case.
• • Check homework or exam work: Type in the coefficients from a homework problem and compare each line to what you wrote, to catch sign errors in b / 2a or the added term.

Completing the square is one of two general methods for solving quadratics; the other is the quadratic formula. The two methods are equivalent because the formula is the final line of completing the square after the algebra has been carried out in full. Running the derivation makes that connection obvious, even for students who have only seen the formula handed to them.

The method shows up in calculus, statistics, and physics whenever a quadratic needs to be reorganised, for example when evaluating an integral, deriving the variance of a probability distribution, or moving a parabola into vertex form.

When you want a randomised set of problems to apply the method to, completing the square practice calculator generates fresh ax² + bx + c = d exercises and walks through each solution.

The calculator divides by a, moves the constant to the right, halves the linear coefficient, squares that value, recognises the perfect square, and takes square roots.

• a: Coefficient of x² in ax² + bx + c = 0. Must be non-zero.
• b: Coefficient of x. Half of b, divided by a, becomes the linear term inside the perfect square.
• c: Constant term. It is moved to the right side before the added term is computed.
• Added term (b / 2a)²: Square of half the linear coefficient. Adding it to both sides turns the left side into a perfect square trinomial.
• Discriminant b² - 4ac: Value under the radical that decides whether the equation has two real, one repeated, or two complex roots.

The same five steps work for every quadratic because (x + b/2a)² = x² + (b/a)x + (b/2a)² always absorbs the linear term once the constant is moved. The discriminant b² - 4ac appears naturally as the numerator of (b² - 4ac) / (4a²).

If the discriminant is positive, the right side is positive and the calculator takes a real square root for two distinct real roots. If it is zero, there is one repeated root at the vertex. If it is negative and the complex-roots toggle is on, the calculator reports h ± bi.

According to Khan Academy, the identity (x + b/2a)² absorbs the linear term of ax² + bx + c for any real coefficients

Once the derivation ends, the final line of the method is the quadratic formula, and quadratic formula calculator lets you skip the algebra and jump straight to the two roots plus the vertex.

Four ideas make the method click: the identity that drives it, the added term, the discriminant sign, and the root split.

Memorise (x + k)² = x² + 2kx + k² first. It is the only algebraic fact that completing the square relies on, and the same identity is reused in calculus.

The added term is the smallest number to add to x² + (b/a)x to make a perfect square. The method is constructive: it tells you which value to add, shows the resulting trinomial, and leaves the rest to the square root. That is why it always works, regardless of which real coefficients you start with.

If the right-hand side value comes back negative, the discriminant is the next thing to inspect, and discriminant calculator classifies the sign of b² - 4ac together with the matching root count.

Type the three coefficients of the quadratic, pick a precision, decide whether to show complex roots, and read the step by step result with the added term, perfect square trinomial, and roots.

1. 1 Write the equation in standard form: Rearrange the quadratic as ax² + bx + c = 0 so the leading coefficient is on the left and the equation equals zero.
2. 2 Enter coefficients a, b, and c: Type the numbers in front of x², x, and the constant. Watch the sign of b (b = -3 in x² - 3x - 4 is entered as -3).
3. 3 Pick a precision: Choose how many decimal places to use. Default 4 is enough.
4. 4 Choose whether to show complex roots: Switch the toggle on to print a ± bi for negative discriminants. Switch it off if you only care about real roots.
5. 5 Read the added term and right side: The added term is (b / 2a)² and the right side is (b² - 4ac) / (4a²). Together they show the perfect square trinomial.
6. 6 Use the roots and vertex to finish: Read the roots, note the discriminant, and combine with the vertex x = -b / 2a to sketch the parabola.

Trying 2x² + 4x - 6 = 0 returns added term 1, right side 4, discriminant 64, real roots x = 1 and x = -3, and vertex x = -1, enough to confirm 2(x + 1)² - 8 = 0.

When the added term makes the left side a perfect square trinomial, factoring trinomials calculator turns the result into a product of two linear factors in one step.

Running the derivation step by step reinforces the algebra, surfaces sign errors quickly, and turns the method into a tool you can reuse for integrals, vertex form, and the quadratic formula.

• • See the method in action: Each step is written out, so the perfect square identity and the added term become concrete instead of abstract.
• • Catch sign errors early: If your paper says b / 2a = +1.5 but the calculator reports -1.5, you know which step to revisit.
• • Learn the quadratic formula naturally: The final line of the derivation is the quadratic formula, so the two methods stop feeling like separate topics.
• • Cover real, repeated, and complex roots: The same five steps work when the discriminant is positive, zero, or negative, which is the whole point of the method.
• • Reuse the result in calculus and physics: Vertex form a(x - h)² + k is the form for integrals, completing the square in probability, or the turning point of a projectile.

Across a homework set, the step by step view is the fastest way to turn a guess into a confirmed answer. The calculator also doubles as a self-check tool: type in textbook coefficients and confirm every line matches the textbook.

After the vertex form a(x - h)² + k is in hand, parabola calculator reads the focus, directrix, and axis of symmetry so the parabola is fully described.

The method always succeeds, but four conditions change how the result looks.

• • This tool accepts real coefficients only. Quadratics with complex coefficients use a different completion and are not covered.
• • The display rounds the added term, right side, discriminant, and roots to the chosen precision. The exact form is preserved in concept but not always in symbolic form on screen.
• • When a is extremely close to zero, the divide-by-a step amplifies rounding errors. Treat the result with caution.

The method always works for any real coefficients a, b, c with a ≠ 0, which is why the verdict at the bottom of the result panel is always the same line. A small change in b or c can flip the discriminant and move the result from two real roots to two complex roots.

According to Cuemath, completing the square always works because the identity (x + b/2a)² = x² + (b/a)x + (b/2a)² is true for every quadratic