EducationMarch 23, 2026

Quadratic Formula: How to Solve Any Quadratic Equation

Q: What is the quadratic formula?

The quadratic formula is x = (-b +/- sqrt(b^2 - 4ac)) / 2a. It solves any equation in the form ax^2 + bx + c = 0, where a is not equal to zero. The formula gives you both solutions (roots) of the equation at once. It was derived from the method of completing the square and works even when factoring is difficult or impossible with rational numbers.

Q: How do you know how many solutions a quadratic equation has?

The discriminant, b^2 - 4ac, tells you. If the discriminant is positive, there are two distinct real solutions. If it equals zero, there is exactly one real solution (a repeated root). If it is negative, there are no real solutions, but two complex (imaginary) solutions. For example, in x^2 + 4x + 4 = 0, the discriminant is 16 - 16 = 0, so there is one repeated root at x = -2.

Q: When should I use the quadratic formula instead of factoring?

Use the quadratic formula when the equation does not factor neatly with integers. For example, x^2 + 3x - 7 = 0 does not factor into nice integer pairs, but the quadratic formula gives exact answers. Factoring is faster when it works (like x^2 + 5x + 6 = (x+2)(x+3)), so try factoring first. If you cannot find integer factors within 30 seconds, switch to the formula. The quadratic formula always works regardless of whether the roots are rational, irrational, or complex.

Q: What are real-world applications of the quadratic formula?

Quadratic equations appear in physics (projectile motion, where h = -16t^2 + vt + h0 gives the height of an object over time), engineering (calculating parabolic structures like bridges and satellite dishes), economics (profit maximization where revenue and cost functions are quadratic), and computer graphics (ray tracing algorithms use quadratic equations to determine where rays intersect curved surfaces). Anytime a quantity depends on the square of another quantity, quadratics are involved.

Q: Can the quadratic formula give complex number answers?

Yes. When the discriminant (b^2 - 4ac) is negative, the square root produces an imaginary number. The solutions take the form x = (-b +/- i*sqrt(|discriminant|)) / 2a, where i is the imaginary unit (the square root of -1). For example, x^2 + 1 = 0 has the discriminant 0 - 4 = -4, giving solutions x = +/- i. Complex roots always come in conjugate pairs: if a + bi is a root, then a - bi is also a root.

By The hakaru Team·Last updated March 2026

The quadratic formula is a universal method for solving equations of the form ax² + bx + c = 0. The formula, x = (-b +/- sqrt(b² - 4ac)) / 2a, returns both solutions of any quadratic equation, whether the roots are rational, irrational, or complex. It is one of the most widely taught formulas in algebra, with roots tracing back over 4,000 years to Babylonian mathematics.

Quick Answer

1. Formula: x = (-b +/- sqrt(b² - 4ac)) / 2a, where a, b, c are coefficients of ax² + bx + c = 0.
2. The discriminant (b² - 4ac) tells you: positive = 2 real roots, zero = 1 root, negative = 2 complex roots.
3. Babylonian mathematicians solved quadratics as early as 2000 BC; the modern formula was published in 1637 by Descartes.
4. Texas Instruments reports that 25 million students worldwide use graphing calculators, many to solve quadratics in algebra classes.

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Understanding the Quadratic Formula

A quadratic equation is any equation that can be written in the standard form ax²+ bx + c = 0, where a, b, and c are constants and a is not zero. The "quadratic" label comes from "quadratus," the Latin word for square, because the variable is squared.

The quadratic formula solves for x directly:

x = (-b +/- sqrt(b² - 4ac)) / 2a

The +/- symbol means you compute two values: one using addition and one using subtraction. These two values are the two roots (solutions) of the equation.

Step-by-Step: How to Use the Quadratic Formula

Let us solve 2x² + 7x - 15 = 0 step by step.

Step 1: Identify a, b, and c

From the equation 2x² + 7x - 15 = 0: a = 2, b = 7, c = -15.

Step 2: Calculate the Discriminant

The discriminant is b² - 4ac = 7² - 4(2)(-15) = 49 + 120 = 169. Since 169 is positive, the equation has two distinct real roots. Since 169 is a perfect square (13²), the roots will be rational numbers.

Step 3: Plug Into the Formula

x = (-7 +/- sqrt(169)) / (2 * 2) = (-7 +/- 13) / 4

Step 4: Calculate Both Roots

x = (-7 + 13) / 4 = 6 / 4 = 3/2 (or 1.5)
x = (-7 - 13) / 4 = -20 / 4 = -5

The solutions are x = 3/2 and x = -5. You can verify by substituting back: 2(1.5)² + 7(1.5) - 15 = 4.5 + 10.5 - 15 = 0.

The Discriminant: How Many Solutions?

The expression under the square root, b² - 4ac, is called the discriminant. It determines the nature and number of solutions without solving the entire equation:

Discriminant Value	Number of Solutions	Nature of Roots	Example
b² - 4ac > 0 (perfect square)	2	Real, rational	x² - 5x + 6 = 0 (D = 1)
b² - 4ac > 0 (not perfect square)	2	Real, irrational	x² - 2x - 1 = 0 (D = 8)
b² - 4ac = 0	1 (repeated)	Real, rational	x² - 6x + 9 = 0 (D = 0)
b² - 4ac < 0	0 real (2 complex)	Complex conjugates	x² + 1 = 0 (D = -4)

Three Methods for Solving Quadratics

Method 1: Factoring

Factoring is the fastest method when it works. To solve x² + 5x + 6 = 0, find two numbers that multiply to 6 and add to 5: those are 2 and 3. So x² + 5x + 6 = (x + 2)(x + 3) = 0, giving x = -2 or x = -3. Factoring only works efficiently when the roots are rational and the coefficients are small integers.

Method 2: Completing the Square

Completing the square transforms the equation into the form (x + h)² = k, which is easy to solve. For x² + 6x + 2 = 0: move the constant (x² + 6x = -2), add (6/2)² = 9 to both sides (x² + 6x + 9 = 7), factor the left side ((x + 3)² = 7), and take the square root (x = -3 +/- sqrt(7)). This method is how the quadratic formula itself is derived.

Method 3: The Quadratic Formula

The quadratic formula works for every quadratic equation, regardless of whether the roots are rational, irrational, or complex. It is the most reliable method, especially when factoring is not obvious. The trade-off is that it requires more arithmetic, but a calculator eliminates that concern.

A Brief History of the Quadratic Formula

The quest to solve quadratic equations spans over 4,000 years of mathematical history:

~2000 BC (Babylon): Babylonian mathematicians solved specific quadratic equations using geometric methods recorded on clay tablets. They could find positive roots through a procedure equivalent to completing the square.
~300 BC (Greece): Euclid developed geometric solutions for quadratics in his work "Elements." The Greek approach was purely geometric, as they had no concept of negative numbers or algebraic notation.
~628 AD (India): Brahmagupta gave the first explicit formula accepting both positive and negative roots. He was the first to recognize that quadratic equations could have two solutions.
~820 AD (Persia): Al-Khwarizmi (whose name gave us the word "algorithm") published systematic methods for solving all types of quadratics, providing both algebraic and geometric proofs.
1637 (France): Rene Descartes published the quadratic formula essentially in its modern form in "La Geometrie."

Real-World Applications

Quadratic equations are not just textbook exercises. They model real phenomena:

Projectile motion: The height of a thrown ball is h(t) = -16t² + v₀t + h₀ (in feet, where v₀ is initial velocity and h₀ is initial height). Solving h(t) = 0 tells you when the ball hits the ground.
Business optimization: If revenue R = -2x² + 200x (where x is units sold), the vertex of the parabola (-b/2a = 50) gives the quantity that maximizes revenue.
Engineering: Parabolic reflectors in satellite dishes and headlights are described by quadratic equations. The focal point of y = ax² is at (0, 1/4a).
Computer graphics: Ray tracing algorithms solve quadratic equations millions of times per frame to determine where light rays intersect curved surfaces.

Common Mistakes to Avoid

Forgetting to set the equation to zero: The formula requires ax² + bx + c = 0. If your equation is x² + 3x = 7, rewrite it as x² + 3x - 7 = 0 before identifying a, b, c.
Wrong sign on b: In -b +/- sqrt(...), the formula negates b. If b = -5, then -b = 5 (positive). Double-check this step.
Dividing by 2 instead of 2a: The denominator is 2a, not just 2. If a = 3, you divide by 6.
Dropping the +/- : The formula produces two answers. Forgetting the minus case means you miss one root.
Arithmetic errors in the discriminant: Be careful with b² - 4ac when c is negative. For example, if c = -3 and a = 2, then -4ac = -4(2)(-3) = +24, not -24.

The Vertex and Axis of Symmetry

Every quadratic equation graphs as a parabola. The vertex (highest or lowest point) is at x = -b / 2a. The y-coordinate of the vertex is found by plugging this x back into the original equation. The axis of symmetry is the vertical line x = -b / 2a. The two roots are always equidistant from the axis of symmetry, which is why the quadratic formula uses -b/2a as the midpoint and adds/subtracts the same value (sqrt(discriminant)/2a) from it.

The Bottom Line

The quadratic formula is the universal tool for solving any equation of the form ax² + bx + c = 0. While factoring is faster when it works and completing the square gives geometric insight, the quadratic formula always delivers both roots. The discriminant (b² - 4ac) is the key shortcut: check it first to know how many real solutions exist before doing the full calculation.

Skip the manual arithmetic and solve any quadratic instantly with our free quadratic formula calculator.

Frequently Asked Questions

What is the quadratic formula?

The quadratic formula is x = (-b +/- sqrt(b^2 - 4ac)) / 2a. It solves any equation in the form ax^2 + bx + c = 0, where a is not equal to zero. The formula gives you both solutions (roots) of the equation at once. It was derived from the method of completing the square and works even when factoring is difficult or impossible with rational numbers.

How do you know how many solutions a quadratic equation has?

The discriminant, b^2 - 4ac, tells you. If the discriminant is positive, there are two distinct real solutions. If it equals zero, there is exactly one real solution (a repeated root). If it is negative, there are no real solutions, but two complex (imaginary) solutions. For example, in x^2 + 4x + 4 = 0, the discriminant is 16 - 16 = 0, so there is one repeated root at x = -2.

When should I use the quadratic formula instead of factoring?

Use the quadratic formula when the equation does not factor neatly with integers. For example, x^2 + 3x - 7 = 0 does not factor into nice integer pairs, but the quadratic formula gives exact answers. Factoring is faster when it works (like x^2 + 5x + 6 = (x+2)(x+3)), so try factoring first. If you cannot find integer factors within 30 seconds, switch to the formula. The quadratic formula always works regardless of whether the roots are rational, irrational, or complex.

What are real-world applications of the quadratic formula?

Quadratic equations appear in physics (projectile motion, where h = -16t^2 + vt + h0 gives the height of an object over time), engineering (calculating parabolic structures like bridges and satellite dishes), economics (profit maximization where revenue and cost functions are quadratic), and computer graphics (ray tracing algorithms use quadratic equations to determine where rays intersect curved surfaces). Anytime a quantity depends on the square of another quantity, quadratics are involved.

Can the quadratic formula give complex number answers?

Yes. When the discriminant (b^2 - 4ac) is negative, the square root produces an imaginary number. The solutions take the form x = (-b +/- i*sqrt(|discriminant|)) / 2a, where i is the imaginary unit (the square root of -1). For example, x^2 + 1 = 0 has the discriminant 0 - 4 = -4, giving solutions x = +/- i. Complex roots always come in conjugate pairs: if a + bi is a root, then a - bi is also a root.

Solve any quadratic equation instantly

Enter the coefficients a, b, and c to get both roots, the discriminant, vertex, and step-by-step solution.

Quadratic Formula Calculator - Free