A quadratic equation is an equation with an x squared term in it — ax squared plus bx plus c equals zero, with a never zero. Its graph is a parabola, a U-shaped curve, and solving the equation means finding where that curve crosses the x-axis. There are four ways to find those crossing points, and they are not rivals: factoring is fastest when it works, the square-root method handles the stripped-down cases, completing the square always works and explains why, and the quadratic formula is completing the square done once, in general, so you never have to repeat the work. This guide works every method on real equations, so you can choose with judgment instead of habit.

What a quadratic equation is

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are numbers and a is not zero. The a is the coefficient of x², the b is the coefficient of x, and the c is the constant — the number standing alone. The condition that a is not zero is what makes the equation quadratic: if a were zero, the x² term would vanish and the equation would be linear, bx + c = 0, with a single solution x = −c ÷ b. Reading a, b and c off an equation is the first skill in the subject, and the most common early error is misreading a sign — in x² − 3x + 2 = 0, b is −3, not 3, and every method below depends on that minus sign.

Graphically, solving means finding the x-intercepts of the parabola y = ax² + bx + c. The curve can cross the x-axis twice, touch it once, or miss it entirely, and those three cases are exactly the three kinds of answer: two distinct real roots, one repeated root, or no real roots. The sign of a tells you which way the parabola opens: positive a opens upward like a cup, so the vertex is a minimum; negative a opens downward, so the vertex is a maximum. The value of c is the y-intercept — the height where the curve crosses the y-axis, at x = 0. Keep the picture in mind while you solve: every algebraic step below is a statement about where this curve meets the axis.

Quadratics also appear in three useful forms. Standard form is ax² + bx + c = 0, the form every method starts from. Factored form is a(x − r₁)(x − r₂) = 0, which displays the roots r₁ and r₂ directly — if either factor is zero, the product is zero, which is the zero-product property that makes factoring work. Vertex form is a(x − h)² + k = 0 in equation terms, or y = a(x − h)² + k as a function, which displays the vertex (h, k) directly. Converting between forms is a skill in itself, covered in the three-forms section below, and the quadratic formula guide treats the formula and the discriminant in full depth.

Factoring: the fastest method, when it works

Factoring rewrites the quadratic as a product of two linear factors and then uses the zero-product property: if a product equals zero, at least one factor equals zero. For x² + 5x + 6 = 0, the factors are (x + 2)(x + 3), because 2 times 3 is 6 and 2 plus 3 is 5. Setting each factor to zero gives x = −2 and x = −3. Checking by substitution: (−2)² + 5(−2) + 6 is 4 − 10 + 6, which is 0; and (−3)² + 5(−3) + 6 is 9 − 15 + 6, which is also 0. The check takes seconds and catches sign errors, which are the usual failure mode — always check, especially when the roots are negative.

When a equals 1, factoring is a search for two numbers whose product is c and whose sum is b. For x² − 3x + 2 = 0, the numbers are −1 and −2: their product is 2 and their sum is −3, giving (x − 1)(x − 2) = 0 and roots 1 and 2. When a is not 1, the search is wider: for 2x² + 7x + 3 = 0, the factors are (2x + 1)(x + 3), because 2x times x is 2x², the outer and inner products 6x and x add to 7x, and 1 times 3 is 3. The roots are x = −1/2 and x = −3. The ac method organises the same search: multiply a by c, find two numbers with that product and sum b, split the middle term, and factor by grouping. It is the same mathematics with bookkeeping attached.

Factoring fails gracefully: some quadratics do not factor over the rational numbers at all. The discriminant tells you in advance — if D = b² − 4ac is a perfect square, the roots are rational and the equation factors neatly; if it is not, factoring by hand will stall, and that is information, not defeat. Switch methods instead of forcing it: completing the square or the formula will finish the equation. A method that tells you when to abandon it is a good method.

The square-root method: the stripped-down cases

When an equation has no x term, or when one side is already a perfect square, the square-root method is the shortest route. For x² − 9 = 0, move the constant over: x² = 9, so x = 3 or x = −3 — both square roots, and forgetting the negative one is the classic error. For (x − 2)² = 25, take the square root of both sides: x − 2 = 5 or x − 2 = −5, giving x = 7 or x = −3. The ± sign is not decoration; it is the second root, and dropping it loses half the answer.

The method also handles equations missing the constant term. For x² − 5x = 0, factor out x: x(x − 5) = 0, giving x = 0 or x = 5. And for an equation like 4x² = 36, divide first: x² = 9, so x = 3 or x = −3. These cases look trivial because they are — the square-root method exists precisely so that simple equations are not solved with heavy machinery. Reach for it whenever the equation is already a square, or becomes one after moving a single term.

Completing the square: the method that always works

Completing the square turns any quadratic into a perfect square plus a constant, which the square-root method then finishes. Take x² + 6x + 5 = 0. Half of 6 is 3, and 3 squared is 9, so x² + 6x + 9 is the perfect square (x + 3)². Rewrite the equation as (x + 3)² − 9 + 5 = 0, which is (x + 3)² = 4. Take square roots: x + 3 = 2 or x + 3 = −2, giving x = −1 or x = −5. Both check: 1 − 6 + 5 is 0, and 25 − 30 + 5 is 0. The method added a number and subtracted it again — nothing changed except the shape, and the new shape exposes the roots.

When a is not 1, divide through by a first. For 2x² + 8x + 6 = 0, dividing by 2 gives x² + 4x + 3 = 0. Half of 4 is 2, and 2 squared is 4, so (x + 2)² − 4 + 3 = 0, which is (x + 2)² = 1, giving x = −1 or x = −3. When the equation has no neat factors, completing the square still proceeds identically — it does not care whether the numbers are friendly. That reliability is why it is taught before the formula: the formula is this exact process carried out once, with letters instead of numbers, so that the result can be reused forever. The derivation is worked in full in the quadratic formula guide.

Completing the square also reads the graph directly. The rewritten form y = (x + 3)² − 4 for x² + 6x + 5 is vertex form: the vertex sits at (−3, −4), the axis of symmetry is x = −3, and the minimum value is −4. One manipulation gives the roots, the vertex, and the axis together. Whenever a question asks for the vertex or the maximum or minimum value rather than the roots, completing the square is usually the intended method.

The quadratic formula in context

The quadratic formula, x = (−b ± √(b² − 4ac)) ÷ 2a, solves any quadratic equation in one substitution. It is not a fifth method competing with the others; it is the general answer that completing the square produces, packaged for reuse. For x² − 3x + 2 = 0, substitute a = 1, b = −3, c = 2: the discriminant is (−3)² − 4 × 1 × 2 = 9 − 8 = 1, and x = (3 ± 1) ÷ 2, giving x = 2 and x = 1. The sign discipline matters most here: b enters the formula negated, so a b of −3 becomes +3, and the discriminant uses b squared, so the sign of b disappears there. Most formula errors are sign errors at exactly these two places.

Use the formula whenever factoring stalls, when the coefficients are fractions or decimals, or when you need the discriminant’s preview of the answer before solving. Its cost is arithmetic care: one substitution, one discriminant, two branches of the ±. Its benefit is certainty — no factoring search, no completing-the-square manipulation, and the same steps every time. The companion guide works the formula’s three cases in full: two real roots, one repeated root, and the complex pair that appears when the discriminant is negative.

Choosing a method: a decision guide

Choose by what the equation looks like, not by habit. If one side is a square or the equation has a missing term, use the square-root method. If the factors are visible — two numbers with product c and sum b, or a perfect-square discriminant — factor. If the question asks for the vertex, the axis, or a maximum or minimum value, complete the square. In every other case, or whenever factoring has stalled after a genuine attempt, use the formula. The table below is the whole decision in one place.

Speed is not the only criterion. On coursework, the method asked for is part of the answer, and showing the requested working matters as much as the roots. On a calculator, the method is invisible and only the roots matter — but knowing which method applies is what lets you check the machine. Work one method, check with another: factor, then verify with the formula’s discriminant, or solve with the formula and check by substituting each root back into the original equation. Two methods agreeing is the strongest evidence an answer is right.

The three forms of a quadratic

Standard form, factored form, and vertex form are the same parabola described for different purposes. Standard form y = ax² + bx + c displays the y-intercept c and feeds every solving method. Factored form y = a(x − r₁)(x − r₂) displays the roots r₁ and r₂: for x² − 3x + 2, the factored form is (x − 1)(x − 2), and the roots 1 and 2 are visible before any solving. Vertex form y = a(x − h)² + k displays the vertex (h, k): for the same equation, the vertex form is (x − 1.5)² − 0.25, so the vertex is (1.5, −0.25) and the minimum value is −0.25.

Converting standard form to factored form is factoring. Converting standard form to vertex form is completing the square: h is −b ÷ 2a, and k is the value of the function at h. For x² + 6x + 5, h is −3 and k is 9 − 18 + 5, which is −4, giving vertex form (x + 3)² − 4. Converting factored form to standard form is expanding. Each conversion is one skill you already own if you can factor and complete the square, and each form answers a different question at a glance. Learn to move between them freely, and the graph, the roots, and the vertex stop being separate topics.

One more conversion is worth knowing: building an equation from its roots. If the roots are r₁ and r₂, the equation is (x − r₁)(x − r₂) = 0, which expands to x² − (r₁ + r₂)x + r₁r₂ = 0. Roots 2 and 1 give x² − 3x + 2 = 0 — the equation this guide has used throughout, now built backwards. The sum of the roots is −b ÷ a and the product is c ÷ a, relations worth checking against any answer: for x² − 3x + 2, the roots 1 and 2 sum to 3, which is −(−3) ÷ 1, and multiply to 2, which is c. If your roots fail either check, an error is hiding somewhere.

Checking your roots

Every root should be checked by substitution: put it back into the original equation and confirm the result is zero. For x² + 5x + 6 = 0 with roots −2 and −3, substitution gives 4 − 10 + 6 = 0 and 9 − 15 + 6 = 0. For the formula’s answer to 2x² − 4x − 6 = 0, roots 3 and −1, substitution gives 18 − 12 − 6 = 0 and 2 + 4 − 6 = 0. Substitution catches sign errors, arithmetic slips, and wrong methods alike, and it costs one line per root.

Two further checks cost nothing. The sum-product relations above must hold: roots sum to −b ÷ a and multiply to c ÷ a. And the graph must agree: two real roots mean the parabola crosses the axis twice, one repeated root means it touches once, and no real roots mean it never touches. If the algebra says two roots but the graph you sketched never crosses, one of the two is wrong — find out which before trusting either. The Quadratic Formula Calculator draws the parabola with the roots marked, which makes the graphical check immediate.

Frequently asked questions

What is a quadratic equation?

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are numbers and a is not zero. The x² term is what makes it quadratic. Its graph is a parabola, and solving it means finding where the parabola crosses the x-axis — the roots. If a were zero, the equation would be linear, not quadratic.

How do you factor a quadratic equation?

Rewrite it as a product of two linear factors, then set each factor to zero. For x² + 5x + 6 = 0, the factors are (x + 2)(x + 3), because 2 × 3 = 6 and 2 + 3 = 5, giving roots −2 and −3. When a = 1, look for two numbers with product c and sum b. When a ≠ 1, use the ac method or factor by grouping. Check each root by substitution.

How do you solve a quadratic by completing the square?

Turn the equation into a perfect square plus a constant, then take square roots. For x² + 6x + 5 = 0: half of 6 is 3, 3² = 9, so (x + 3)² − 9 + 5 = 0, giving (x + 3)² = 4, so x + 3 = ±2 and x = −1 or x = −5. If a ≠ 1, divide through by a first. The method always works, even when factoring does not.

When should you use the quadratic formula instead of factoring?

Use the formula when factoring stalls, when the discriminant is not a perfect square, or when the coefficients are fractions or decimals. The formula x = (−b ± √(b² − 4ac)) ÷ 2a solves any genuine quadratic in one substitution. Factoring is faster when the factors are visible; the formula is the reliable fallback that never fails on a quadratic.

What does the discriminant tell you?

The discriminant D = b² − 4ac previews the answer before you solve. If D is positive, there are two distinct real roots. If D is zero, there is one repeated root. If D is negative, there are no real roots — the solutions are a complex conjugate pair. If D is a perfect square (with rational coefficients), the roots are rational and the equation factors neatly.

How do you find the vertex of a parabola?

The vertex sits on the axis of symmetry at x = −b ÷ 2a. Substitute that x back into the equation for the y-coordinate, or complete the square to vertex form y = a(x − h)² + k, where the vertex is (h, k). For x² − 3x + 2, the vertex is (1.5, −0.25), a minimum because a is positive.