Solve any quadratic equation ax² + bx + c = 0 instantly. Get real and complex roots with discriminant analysis and step-by-step solution.
Enter the coefficients a, b, and c for the equation ax² + bx + c = 0.
The quadratic formula is a mathematical formula that provides the solution(s) to any quadratic equation of the form ax² + bx + c = 0, where a ≠ 0. It is one of the most important formulas in algebra, used by students, engineers, scientists, and economists worldwide.
This formula was known to ancient Babylonian mathematicians as far back as 2000 BCE, though in a different form. The modern algebraic notation was developed during the Renaissance. Khan Academy's algebra course provides excellent video lessons on understanding and applying the quadratic formula.
The discriminant (b² - 4ac) is the key to understanding the nature of a quadratic equation's solutions:
The discriminant concept extends to higher-degree polynomials and is fundamental in algebraic number theory.
Solve: 2x² - 7x + 3 = 0
Identify: a = 2, b = -7, c = 3
Discriminant: (-7)² - 4(2)(3) = 49 - 24 = 25
Square root: √25 = 5
Solutions:
x₁ = (7 + 5) / 4 = 12/4 = 3
x₂ = (7 - 5) / 4 = 2/4 = 0.5
When a quadratic can be factored, it's often the fastest method. For example, x² - 5x + 6 = (x-2)(x-3) = 0 gives x = 2 or x = 3. Finding factors often relies on the Greatest Common Factor.
This method transforms the equation into a perfect square trinomial. It's actually how the quadratic formula itself is derived, as explained in Purplemath's completing the square guide.
The solutions (roots) of a quadratic equation are the x-intercepts of the corresponding parabola y = ax² + bx + c.
If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The solution is simply x = -c/b. A quadratic equation must have a ≠ 0.
When the discriminant is negative, the square root of a negative number involves the imaginary unit i (where i² = -1). The roots come in conjugate pairs: a + bi and a - bi. Complex numbers are widely used in electrical engineering and quantum physics.
The quadratic formula is derived by completing the square on the general equation ax² + bx + c = 0. You isolate x by systematically transforming the equation into a perfect square on one side.
No, the quadratic formula only works for degree-2 equations. Cubic (degree 3) and quartic (degree 4) have their own formulas. Equations of degree 5 or higher have no general algebraic formula (proven by Abel-Ruffini theorem).
The quadratic formula solves any equation of the form ax² + bx + c = 0, where a is not zero. It is one of the few tools in school mathematics that works universally: every quadratic equation yields to it, including those that cannot be factored by inspection. The formula also reveals, through its discriminant, how many real solutions exist before you finish calculating them.
The expression under the square root, b² - 4ac, is called the discriminant. It determines the character of the solutions on its own.
| Discriminant | Solutions | Graph Meaning |
|---|---|---|
| Greater than 0 | Two distinct real solutions | Parabola crosses the x-axis twice |
| Equal to 0 | One repeated real solution | Parabola touches the x-axis once |
| Less than 0 | No real solutions | Parabola never reaches the x-axis |
Step 1: Identify the coefficients
Step 2: Calculate the discriminant
Step 3: Apply the formula
Step 4: Split into two solutions
Step 5: Check by substitution
The solutions are x = 3 and x = 0.5
The formula begins with negative b. If b is already negative, as in the example above, negative b becomes positive. Sign errors at this step are the single most common source of wrong answers. Write out the substitution explicitly rather than doing it mentally.
The formula requires ax² + bx + c = 0. Equations often arrive in other arrangements and must be rearranged first, with every term moved to one side.
Only now can the formula be applied
Quadratic equations describe any situation with a squared relationship, which turns out to be a great many. Projectile motion under gravity follows a parabola, which is why the formula appears throughout physics and engineering. Profit maximisation problems in economics, area optimisation in construction, and parabolic reflector design in optics all reduce to solving quadratics.
The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0. It is not an arbitrary rule but the algebraic consequence of that process, which is why it works for every quadratic without exception. Methods for solving quadratics appear in Babylonian tablets from around 2000 BC, long before the modern notation existed.
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