What is the quadratic formula?

The quadratic formula solves any equation of the form ax² + bx + c = 0. The two solutions are x = (-b ± √(b²-4ac)) / (2a). The ± gives two roots: x₁ = (-b + √(b²-4ac)) / (2a) and x₂ = (-b - √(b²-4ac)) / (2a). When the discriminant b²-4ac is positive, both roots are real and distinct. When it equals zero, there is one repeated real root. When negative, the roots are complex conjugates.

What is the discriminant?

The discriminant is the expression b² - 4ac from under the square root in the quadratic formula. It determines the nature of the roots without solving the equation. Discriminant > 0: two distinct real roots. Discriminant = 0: one repeated real root (the parabola touches the x-axis at one point). Discriminant < 0: two complex conjugate roots (the parabola does not cross the x-axis).

What is the vertex of a parabola?

The vertex is the highest or lowest point of the parabola. Its x-coordinate is -b/(2a), the axis of symmetry. Its y-coordinate is found by substituting this x back into the equation: y = c - b²/(4a). If a > 0, the parabola opens upward and the vertex is the minimum. If a < 0, it opens downward and the vertex is the maximum.

What is completing the square?

Completing the square transforms ax² + bx + c into the vertex form a(x - h)² + k, where (h, k) is the vertex. The process: divide by a, move c/a to the right, add (b/2a)² to both sides, factor the left side as a perfect square trinomial, then solve. The quadratic formula is derived by completing the square on the general equation ax² + bx + c = 0.

When does a quadratic have complex roots?

A quadratic equation has complex roots when its discriminant b² - 4ac is negative. The square root of a negative number is imaginary (involving i = √(-1)), so the roots take the form x = (-b ± i√(4ac-b²)) / (2a). Complex roots of a quadratic always come in conjugate pairs: if p + qi is a root, then p - qi is also a root.

Quadratic Formula Calculator

Solve ax² + bx + c = 0 · Real and complex roots · Vertex · Graph · Step-by-step

1x² + 0x + 0 = 0

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The Quadratic Formula

A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. The quadratic formula gives both solutions directly from the coefficients:

x = (-b ± √(b² - 4ac)) / (2a) x₁ = (-b + √(b²-4ac)) / (2a) x₂ = (-b - √(b²-4ac)) / (2a)

Derivation by Completing the Square

The quadratic formula is derived by completing the square on the general equation:

ax² + bx + c = 0 x² + (b/a)x + c/a = 0 (divide by a) x² + (b/a)x = -c/a (rearrange) x² + (b/a)x + (b/2a)² = -c/a + (b/2a)² (complete) (x + b/2a)² = (b²-4ac) / 4a² (factor) x + b/2a = ±√(b²-4ac) / 2a x = (-b ± √(b²-4ac)) / 2a (solve)

The Discriminant

The discriminant Δ = b² - 4ac determines the nature of the roots before solving:

Discriminant	Nature of Roots	Parabola
Δ > 0	Two distinct real roots	Crosses x-axis twice
Δ = 0	One repeated real root	Touches x-axis once (tangent)
Δ < 0	Two complex conjugate roots	Does not cross x-axis

Example 1: x²-5x+6=0 · Δ = 25-24 = 1 > 0 → two real roots Example 2: x²-6x+9=0 · Δ = 36-36 = 0 → one repeated root (x=3) Example 3: x²+2x+5=0 · Δ = 4-20 = -16 < 0 → complex roots

Vieta's Formulas

For any quadratic ax² + bx + c = 0 with roots x₁ and x₂:

Sum of roots: x₁ + x₂ = -b/a Product of roots: x₁ × x₂ = c/a These hold for both real and complex roots.

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Vieta's formulas let you quickly check whether computed roots are correct. For x² - 3x + 2 = 0 with roots 1 and 2: sum = 1 + 2 = 3 = -(-3)/1 ✓, product = 1 × 2 = 2 = 2/1 ✓.

The Parabola: Vertex, Axis of Symmetry, and Forms

Every quadratic y = ax² + bx + c graphs as a parabola. Understanding its geometric properties links the algebraic and visual representations.

Vertex

The vertex is the turning point of the parabola - the minimum point when a > 0, the maximum when a < 0. Its coordinates are:

x-coordinate of vertex: h = -b / (2a) y-coordinate of vertex: k = c - b² / (4a) Also: k = f(h) = a·h² + b·h + c

The Three Forms of a Quadratic

Form	Equation	Best Used For
Standard	ax² + bx + c	Identifying a, b, c; applying quadratic formula
Factored	a(x - r₁)(x - r₂)	Reading x-intercepts (roots) directly
Vertex	a(x - h)² + k	Reading vertex (h, k) and axis of symmetry x = h

Converting from standard to vertex form completes the square. For x² - 3x + 2:

x² - 3x + 2 = (x² - 3x + 9/4) + 2 - 9/4 = (x - 3/2)² - 1/4 Vertex: (1.5, -0.25) · Axis: x = 1.5

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The axis of symmetry always passes through the vertex and is perpendicular to the x-axis: x = h = -b/(2a). The two roots (if real) are always equidistant from the axis of symmetry. The midpoint of the two roots equals the x-coordinate of the vertex.

Complex Roots

When the discriminant is negative, the square root of a negative number appears. The imaginary unit i is defined as i = √(-1), so √(-k) = i√k for any positive k.

x = (-b ± i√(4ac-b²)) / (2a) Example: x² + 2x + 5 = 0 Δ = 4 - 20 = -16 x = (-2 ± √(-16)) / 2 = (-2 ± 4i) / 2 x₁ = -1 + 2i · x₂ = -1 - 2i

Complex roots always appear as conjugate pairs: if p + qi is a root, then p - qi is also a root. The real part p = -b/(2a) is the x-coordinate of the vertex, and qi represents the imaginary distance the parabola "misses" the x-axis by.

On the graph, a parabola with complex roots lies entirely above (if a > 0) or entirely below (if a < 0) the x-axis. The vertex is the point closest to the x-axis.

Frequently Asked Questions

What does it mean when a quadratic has a repeated root?

A repeated root (also called a double root) occurs when the discriminant equals zero. The quadratic formula gives x = -b/(2a) for both solutions, so there is only one distinct x-value. The parabola touches the x-axis at exactly one point - the vertex - and is tangent to the x-axis there. Example: x² - 6x + 9 = 0 = (x - 3)², so x = 3 is a repeated root.

Can the quadratic formula handle all quadratic equations?

Yes, the quadratic formula works for any equation of the form ax² + bx + c = 0 where a ≠ 0 and a, b, c are real numbers. It produces real roots when Δ ≥ 0 and complex roots when Δ < 0. The only restriction is a ≠ 0 - if a = 0, the equation is linear (bx + c = 0), not quadratic, and the formula produces a division by zero.

How do you factor a quadratic using its roots?

Once you find the roots x₁ and x₂, the factored form is a(x - x₁)(x - x₂). Example: 2x² - 8x + 6. Divide by 2: x² - 4x + 3. Roots: (4 ± √(16-12))/2 = (4 ± 2)/2 = 3 or 1. Factored: 2(x - 3)(x - 1). Verify: 2(x² - 4x + 3) = 2x² - 8x + 6 ✓. For complex roots, the factored form involves complex numbers.

What are some real-world applications of quadratic equations?

Projectile motion: the height of a thrown ball follows h(t) = -½gt² + v₀t + h₀, a quadratic in t. Setting h(t) = 0 finds when the ball lands. Area problems: a rectangular field with given perimeter and area constraints yields a quadratic. Optics: the thin lens equation can be rearranged into quadratic form. Engineering: beam deflection, circuit analysis (LC circuits), and structural load calculations all produce quadratic equations that are solved with the formula.