What is the quadratic formula and how does it work?

The quadratic formula x = (-b ± sqrt(b²-4ac)) / (2a) gives the exact roots of any equation ax² + bx + c = 0. It is derived by completing the square on the standard form. The formula handles all cases: rational roots, irrational roots, and complex roots, making it the universal method for solving quadratics.

What is the discriminant of a quadratic equation?

The discriminant D = b² - 4ac determines the nature of the roots without fully solving. D > 0 gives two distinct real roots. D = 0 gives one repeated real root (the parabola touches the x-axis at exactly one point). D < 0 gives two complex conjugate roots with no real x-intercepts.

How do I identify a, b, and c in a quadratic equation?

Rewrite the equation in standard form ax² + bx + c = 0. The coefficient of x² is a, the coefficient of x is b, and the constant term is c. For 3x² - 7x + 2 = 0, a = 3, b = -7, c = 2. Always include the sign: if the x term is subtracted, b is negative.

What does a repeated root mean geometrically?

A repeated root (D = 0) means the parabola y = ax² + bx + c is tangent to the x-axis at exactly one point. The single root x = -b/(2a) is also the x-coordinate of the vertex. The parabola touches zero without crossing it.

What are complex conjugate roots?

When D < 0, the roots are complex numbers of the form p + qi and p - qi, where i = sqrt(-1). They always come in conjugate pairs for equations with real coefficients. Complex roots mean the parabola does not intersect the x-axis at all. The real part p = -b/(2a) is the axis of symmetry.

How do I verify my quadratic solutions?

Substitute each root back into ax² + bx + c. The result should equal zero. For example, if x = 3 is a root of x² - 5x + 6 = 0, check: 9 - 15 + 6 = 0. You can also use Vieta's formulas: the sum of roots equals -b/a and the product of roots equals c/a.

What are Vieta's formulas for a quadratic equation?

For ax² + bx + c = 0 with roots x₁ and x₂: sum of roots x₁ + x₂ = -b/a, and product of roots x₁ × x₂ = c/a. For x² - 5x + 6 = 0: sum = 3 + 2 = 5 = -(-5)/1, product = 3 × 2 = 6 = 6/1. Vieta's formulas let you verify roots quickly and construct quadratics from known roots.

What is the vertex form of a quadratic?

Vertex form is a(x - h)² + k, where (h, k) is the vertex of the parabola. The vertex x-coordinate is h = -b/(2a) and the y-coordinate is k = c - b²/(4a). Vertex form makes the maximum or minimum value of the parabola immediately visible: k is the extreme value, attained at x = h.

Can all quadratic equations be solved with the quadratic formula?

Yes. The quadratic formula works for every quadratic ax² + bx + c = 0 with a not equal to zero, regardless of whether the coefficients are integers, fractions, or irrational numbers. Other methods like factoring or completing the square are faster in special cases, but the quadratic formula is always applicable.

What is the axis of symmetry of a parabola?

The axis of symmetry is the vertical line x = -b/(2a) that passes through the vertex and divides the parabola into two mirror-image halves. It is also exactly halfway between the two real roots. For x² - 5x + 6 = 0 with roots 2 and 3, the axis of symmetry is x = 2.5.

Quadratic Formula Calculator

Q: How do I identify a, b, and c in a quadratic equation?

Rewrite the equation in standard form ax² + bx + c = 0. The coefficient of x² is a, the coefficient of x is b, and the constant term is c. For 3x² - 7x + 2 = 0, a = 3, b = -7, c = 2. Always include the sign: if the x term is subtracted, b is negative.

Q: What does a repeated root mean geometrically?

A repeated root (D = 0) means the parabola y = ax² + bx + c is tangent to the x-axis at exactly one point. The single root x = -b/(2a) is also the x-coordinate of the vertex. The parabola touches zero without crossing it.

Q: What are complex conjugate roots?

When D < 0, the roots are complex numbers of the form p + qi and p - qi, where i = sqrt(-1). They always come in conjugate pairs for equations with real coefficients. Complex roots mean the parabola does not intersect the x-axis at all. The real part p = -b/(2a) is the axis of symmetry.

Q: How do I verify my quadratic solutions?

Substitute each root back into ax² + bx + c. The result should equal zero. For example, if x = 3 is a root of x² - 5x + 6 = 0, check: 9 - 15 + 6 = 0. You can also use Vieta's formulas: the sum of roots equals -b/a and the product of roots equals c/a.

Q: What are Vieta's formulas for a quadratic equation?

For ax² + bx + c = 0 with roots x₁ and x₂: sum of roots x₁ + x₂ = -b/a, and product of roots x₁ × x₂ = c/a. For x² - 5x + 6 = 0: sum = 3 + 2 = 5 = -(-5)/1, product = 3 × 2 = 6 = 6/1. Vieta's formulas let you verify roots quickly and construct quadratics from known roots.

Q: What is the vertex form of a quadratic?

Vertex form is a(x - h)² + k, where (h, k) is the vertex of the parabola. The vertex x-coordinate is h = -b/(2a) and the y-coordinate is k = c - b²/(4a). Vertex form makes the maximum or minimum value of the parabola immediately visible: k is the extreme value, attained at x = h.

Q: Can all quadratic equations be solved with the quadratic formula?

Yes. The quadratic formula works for every quadratic ax² + bx + c = 0 with a not equal to zero, regardless of whether the coefficients are integers, fractions, or irrational numbers. Other methods like factoring or completing the square are faster in special cases, but the quadratic formula is always applicable.

Q: What is the axis of symmetry of a parabola?

The axis of symmetry is the vertical line x = -b/(2a) that passes through the vertex and divides the parabola into two mirror-image halves. It is also exactly halfway between the two real roots. For x² - 5x + 6 = 0 with roots 2 and 3, the axis of symmetry is x = 2.5.

Enter coefficients a, b, and c to solve ax² + bx + c = 0. Instant roots with discriminant, vertex, and step-by-step working.

📐 What is the Quadratic Formula?

The quadratic formula is the universal algebraic method for finding the roots (solutions) of any quadratic equation in the standard form ax² + bx + c = 0, where a, b, and c are real numbers and a is not equal to zero. The formula is: x = (-b ± sqrt(b² - 4ac)) / (2a). It was derived from the technique of completing the square and works for every possible quadratic, whether the roots are integers, fractions, irrational surds, or complex numbers.

The quadratic formula has direct real-world applications across many fields. Engineers use it to calculate projectile trajectories, optimize structural load distributions, and solve resonance frequencies in circuits. Financial analysts apply quadratic equations to break-even analysis, profit maximization, and bond pricing. Physicists use it for kinematics problems involving constant acceleration, while computer graphics programmers rely on it for ray-sphere intersection tests, bezier curve calculations, and collision detection algorithms. Any time a quantity grows or shrinks in proportion to the square of another variable, a quadratic equation arises.

A critical part of the quadratic formula is the expression under the square root sign, called the discriminant: D = b² - 4ac. The discriminant reveals the nature of the solutions before you even finish the calculation. A positive discriminant produces two distinct real roots, corresponding to two x-intercepts on the parabola y = ax² + bx + c. A discriminant of exactly zero produces one repeated real root, meaning the parabola just touches the x-axis at its vertex. A negative discriminant means the parabola does not cross the x-axis at all, and the two roots are complex conjugate numbers involving i = sqrt(-1).

This calculator solves ax² + bx + c = 0 instantly by applying the quadratic formula step by step. It displays both roots, the discriminant and its interpretation, the vertex of the parabola, and the axis of symmetry. The step-by-step panel shows every arithmetic substitution, making it suitable for checking homework, understanding the method, or verifying manual calculations in engineering and science contexts.

📐 Formula

x = (−b ± √(b² − 4ac)) ÷ (2a)

a = coefficient of x² (must be non-zero)

b = coefficient of x

c = constant term

D = b² − 4ac = discriminant (determines nature of roots)

Vertex: h = −b/(2a), k = c − b²/(4a)

Example: For x² − 5x + 6 = 0, a=1, b=−5, c=6, D = 25−24 = 1, roots = (5 ± 1)/2 = 3 and 2

📖 How to Use This Calculator

Steps

Write the equation in standard form: rearrange so all terms are on one side: ax² + bx + c = 0. Identify a, b, and c, including their signs.

Enter coefficients a, b, and c: type each coefficient into the corresponding field. Use negative values where needed (e.g. b = -5 if the equation has -5x).

Click Solve: the calculator applies the quadratic formula and instantly shows the discriminant, both roots (real or complex), vertex, and axis of symmetry.

Read the step-by-step working: the working panel shows every substitution step explicitly, from discriminant calculation through to the final root values.

Verify with Vieta's formulas: confirm correctness: sum of roots should equal -b/a and product of roots should equal c/a.

💡 Example Calculations

Example 1: Two Distinct Real Roots

Solve x² - 5x + 6 = 0 (a=1, b=-5, c=6)

Discriminant: D = (-5)² - 4(1)(6) = 25 - 24 = 1

D = 1 > 0, so two distinct real roots exist.

x₁ = (-(-5) + sqrt(1)) / (2×1) = (5 + 1) / 2 = 3

x₂ = (-(-5) - sqrt(1)) / (2×1) = (5 - 1) / 2 = 2

Roots: x₁ = 3, x₂ = 2 (Vertex at (2.5, -0.25))

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Example 2: Repeated Root (D = 0)

Solve x² - 6x + 9 = 0 (a=1, b=-6, c=9)

Discriminant: D = (-6)² - 4(1)(9) = 36 - 36 = 0

D = 0, so exactly one repeated real root (parabola tangent to x-axis).

x = -b / (2a) = 6 / 2 = 3

Root: x = 3 (repeated), which is also x² - 6x + 9 = (x - 3)²

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Example 3: Complex Conjugate Roots (D < 0)

Solve x² + 2x + 5 = 0 (a=1, b=2, c=5)

Discriminant: D = (2)² - 4(1)(5) = 4 - 20 = -16

D = -16 < 0, so two complex conjugate roots (no real x-intercepts).

Real part: -b/(2a) = -2/2 = -1. Imaginary part: sqrt(16)/(2×1) = 4/2 = 2.

Roots: x₁ = -1 + 2i, x₂ = -1 - 2i

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Example 4: Irrational Roots

Solve 2x² - 3x - 2 = 0 (a=2, b=-3, c=-2)

Discriminant: D = (-3)² - 4(2)(-2) = 9 + 16 = 25

x₁ = (3 + sqrt(25)) / (2×2) = (3 + 5) / 4 = 8/4 = 2

x₂ = (3 - sqrt(25)) / (2×2) = (3 - 5) / 4 = -2/4 = -0.5

Roots: x₁ = 2, x₂ = -0.5. Verify: sum = 1.5 = 3/2 = -(-3)/2 ✓, product = -1 = -2/2 ✓

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❓ Frequently Asked Questions

What is the quadratic formula and when should I use it?+

The quadratic formula x = (-b ± sqrt(b² - 4ac)) / (2a) solves any equation of the form ax² + bx + c = 0. Use it when the equation does not factor easily or when you need an exact numerical answer. It works for all cases including irrational and complex roots, making it the most reliable single method for quadratic equations.

How do I put a quadratic equation into standard form?+

Move all terms to one side so the equation reads ax² + bx + c = 0. For example, 3x² = 7x - 2 becomes 3x² - 7x + 2 = 0 by subtracting 7x and adding 2. Then read off a = 3, b = -7, c = 2. Always include the sign of each coefficient in the value you enter.

What does D greater than zero mean for the roots?+

When D = b² - 4ac is positive, the equation has two distinct real roots. The square root of a positive number is a real number, so both x₁ = (-b + sqrt(D))/(2a) and x₂ = (-b - sqrt(D))/(2a) are real and unequal. On the parabola graph, these are the two x-intercepts where the curve crosses the horizontal axis.

What does D equal to zero mean?+

When D = 0, the formula gives x = -b/(2a) with the plus and minus both yielding the same value. This is called a repeated or double root. Geometrically, the parabola is tangent to the x-axis at exactly one point, the vertex. The equation factors as a(x - r)² = 0 for this root r.

What does D less than zero mean and what are complex roots?+

When D is negative, sqrt(D) involves sqrt(-1) = i, producing two complex conjugate roots a + bi and a - bi. The parabola does not intersect the x-axis at all. Complex roots always appear in conjugate pairs when the original coefficients are real numbers. They are valid mathematical solutions even though they are not real numbers.

Can the quadratic formula handle non-integer coefficients?+

Yes. The quadratic formula works for any real numbers a, b, and c, including decimals and fractions. For example, 0.5x² + 1.5x - 1 = 0 uses a = 0.5, b = 1.5, c = -1. The formula will return exact decimal values for the roots. Enter decimals directly into the calculator fields.

What is the vertex of a parabola and how is it calculated?+

The vertex is the highest or lowest point on the parabola y = ax² + bx + c. Its x-coordinate is h = -b/(2a), halfway between the two roots (when real). Its y-coordinate is k = c - b²/(4a), the minimum value when a > 0 or maximum when a < 0. The vertex form is a(x - h)² + k.

How does the quadratic formula relate to factoring?+

If the roots are x₁ and x₂, the quadratic factors as a(x - x₁)(x - x₂). For x² - 5x + 6 = 0 with roots 3 and 2, the factored form is (x - 3)(x - 2). Factoring is faster when roots are simple integers, but the quadratic formula always works even when factoring by inspection is not possible.

How can I verify the roots found by the quadratic formula?+

Two quick checks: (1) Substitute each root back into the original equation to confirm both sides equal zero. (2) Use Vieta's formulas: the sum of roots should equal -b/a and the product should equal c/a. For example, roots 3 and 2 for x² - 5x + 6 = 0: sum = 5 = -(-5)/1, product = 6 = 6/1.

What is the axis of symmetry of a quadratic parabola?+

The axis of symmetry is the vertical line x = -b/(2a) that divides the parabola into two mirror halves. When the equation has two real roots, the axis of symmetry passes exactly halfway between them. It is also the x-coordinate of the vertex. For x² - 5x + 6 = 0, axis = x = 2.5, midpoint of roots 2 and 3.

What is the difference between roots, zeros, and solutions of a quadratic?+

These terms describe the same values. "Roots" is the algebraic term (roots of the equation), "zeros" is the function term (zeros of f(x) = ax² + bx + c, where the function equals zero), and "solutions" is the equation-solving term. All three refer to the x-values that satisfy ax² + bx + c = 0.

Why must the coefficient a be non-zero in a quadratic equation?+

When a = 0, the x² term vanishes and the equation becomes bx + c = 0, which is linear, not quadratic. Dividing by 2a in the quadratic formula would require dividing by zero, which is undefined. A quadratic equation by definition has a non-zero leading coefficient to ensure the degree-2 term exists and the parabolic shape is present.

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📌 Quick Tips

💡The discriminant D = b² - 4ac tells you the root type before solving: D > 0 means two real roots, D = 0 means one repeated root, D < 0 means two complex roots.

💡Rewrite your equation in standard form ax² + bx + c = 0 before entering coefficients. Move all terms to one side first.

💡To check your roots, substitute each back into the original equation. Both sides should equal zero.

💡The vertex x-coordinate h = -b/(2a) is exactly halfway between the two roots. You can find the vertex without using the discriminant.

💡For complex roots, the parabola does not cross the x-axis. The vertex is above the x-axis when a > 0, or below it when a < 0.

Quadratic Formula Calculator

Step-by-Step Working

📐 What is the Quadratic Formula?

📐 Formula

📖 How to Use This Calculator

Steps

💡 Example Calculations

Example 1: Two Distinct Real Roots

Solve x² - 5x + 6 = 0 (a=1, b=-5, c=6)

Example 2: Repeated Root (D = 0)

Solve x² - 6x + 9 = 0 (a=1, b=-6, c=9)

Example 3: Complex Conjugate Roots (D < 0)

Solve x² + 2x + 5 = 0 (a=1, b=2, c=5)

Example 4: Irrational Roots

Solve 2x² - 3x - 2 = 0 (a=2, b=-3, c=-2)

❓ Frequently Asked Questions

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📌 Quick Tips