Can the quadratic formula solve all quadratic equations?

Yes — it works for any ax² + bx + c = 0 where a ≠ 0, including equations with no real roots. When the discriminant is negative, it produces complex roots in a + bi form.

What does it mean when the discriminant is negative?

A negative discriminant means the equation has no real solutions. The parabola sits entirely above or below the x-axis. The solutions are complex numbers.

How is the quadratic formula derived?

The formula comes from completing the square on ax² + bx + c = 0. You isolate the squared term, complete the square, take the square root of both sides, and simplify.

What is the connection between the quadratic formula and the parabola?

The roots are the x-intercepts of the parabola y = ax² + bx + c. The vertex x-coordinate is always -b/(2a) — the midpoint between the two roots.

Quadratic Formula Calculator

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Enter your values and press Calculate — your answer, visual and step-by-step solution appear here.

What Is the Quadratic Formula?

The quadratic formula solves any equation of the form ax² + bx + c = 0 for x. It gives you the roots directly — no guessing, no factoring trial and error.

x = (−b ± √(b² − 4ac)) / 2a

The ± symbol means there are two calculations: one with addition and one with subtraction. That's why most quadratic equations have two solutions.

Variable	Role in ax² + bx + c = 0	Example: 2x² + 5x − 3 = 0
a	Coefficient of x²	a = 2
b	Coefficient of x	b = 5
c	Constant term	c = −3

How to Use This Calculator

Enter a — the coefficient in front of x².
Enter b — the coefficient in front of x. Include the sign (negative b is common).
Enter c — the constant term. Use 0 if there is no constant.
Click Calculate — you get both roots, the discriminant, a graph, and a full step-by-step solution. No login required.

Tip: Use the Hint button to work through the problem yourself before revealing the full solution — great for exam prep.

Understanding the Discriminant (b² − 4ac)

The discriminant is the expression under the square root: b² − 4ac. Before you even finish solving, it tells you how many real solutions the equation has.

Discriminant Value	Number of Solutions	What It Means
b² − 4ac > 0	2 real solutions	The parabola crosses the x-axis at two points
b² − 4ac = 0	1 real solution (repeated)	The parabola just touches the x-axis at one point
b² − 4ac < 0	No real solutions	The parabola sits entirely above or below the x-axis

When the discriminant is negative, the solutions are complex numbers in the form a + bi. They are mathematically valid — just not visible on the real number line.

Step-by-Step Worked Examples

Example 1 — Two Real Solutions: 2x² + 5x − 3 = 0

Identify a, b, c. a = 2, b = 5, c = −3.
Calculate the discriminant. b² − 4ac = 25 − 4(2)(−3) = 25 + 24 = 49.
Take the square root. √49 = 7.
Apply the ± formula. x = (−5 + 7) / 4 = 0.5 and x = (−5 − 7) / 4 = −3.
Solutions: x = 0.5 and x = −3.

Watch out: b = 5, so −b = −5. If b had been negative (say b = −5), then −b = +5. Always write out −b explicitly to avoid sign errors.

Example 2 — One Repeated Solution: x² − 6x + 9 = 0

Identify a, b, c. a = 1, b = −6, c = 9.
Discriminant. (−6)² − 4(1)(9) = 36 − 36 = 0.
Single solution. x = −(−6) / (2 × 1) = 6/2 = 3.
The parabola touches the x-axis exactly once at x = 3.

Example 3 — No Real Solutions (Complex Roots): x² + 2x + 5 = 0

Identify a, b, c. a = 1, b = 2, c = 5.
Discriminant. 4 − 20 = −16. Negative — no real solutions.
Complex solutions. x = (−2 ± √(−16)) / 2 = (−2 ± 4i) / 2.
Simplified: x = −1 + 2i and x = −1 − 2i.

Writing complex roots: √(−16) = 4i because i = √(−1). Always write the answer in a + bi form: separate the real part (−1) from the imaginary part (±2i).

When to Use the Quadratic Formula

There are three methods for solving quadratic equations. Here's when to use each:

Method	Use When	Always Works?
Factoring	The equation factors to nice integers easily	No
Completing the square	You need vertex form or are deriving a formula	Yes, but slow
Quadratic formula	Factoring isn't obvious; coefficients are decimals or fractions	Yes

Rule of thumb: If you can spot factors quickly, factor first. When in doubt, the quadratic formula always works — for any a, b, and c (as long as a ≠ 0).

Common Mistakes to Avoid

Mistake 1 — Forgetting the ±The formula produces two answers: one with + and one with −. Writing only one means you've missed half the solution.

Mistake 2 — Sign error on −bIf b = −5, then −b = +5. Always substitute −b, not b. Writing it out explicitly prevents this error.

Mistake 3 — Dividing only part of the numerator by 2aThe entire expression (−b ± √(b² − 4ac)) must be divided by 2a. Use parentheses to keep it grouped.

Mistake 4 — Not simplifying the square rootIf the discriminant is a perfect square (e.g. 49), simplify √49 = 7 before continuing. Leaving it as √49 in the final answer is not fully simplified.

Mistake 5 — Using a = 0If a = 0, the equation is linear, not quadratic. The quadratic formula requires a ≠ 0 — division by 2a would be division by zero.

Frequently Asked Questions

What is the quadratic formula?

The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. It solves any equation in the form ax² + bx + c = 0 for x, giving both roots directly.
Can the quadratic formula solve all quadratic equations?

Yes — it works for any ax² + bx + c = 0 where a ≠ 0, including equations with no real roots. When the discriminant is negative, it produces complex (imaginary) roots in a + bi form.
What does it mean when the discriminant is negative?

A negative discriminant means the equation has no real solutions. The parabola sits entirely above or below the x-axis and never crosses it. The solutions are complex numbers.
How is the quadratic formula derived?

The formula comes from completing the square on the general equation ax² + bx + c = 0. You isolate x² + (b/a)x, add (b/2a)² to both sides, then take the square root of both sides and simplify.
What is the connection between the quadratic formula and the parabola?

The roots found by the formula are the x-intercepts of the parabola y = ax² + bx + c. The vertex x-coordinate is always −b/(2a) — the midpoint between the two roots.

Not sure whether to factor or use the formula? Read quadratic formula vs factoring for a quick way to decide.