Quadratic Formula کیلکولیٹر

تیز اور درست نتائج کے لیے Quadratic Formula کیلکولیٹر استعمال کریں۔

اس کیلکولیٹر کو کیسے استعمال کریں

a (coefficient of x²) درج کریں
b (coefficient of x) درج کریں
c (constant) درج کریں
حساب کریں بٹن پر کلک کریں
کیلکولیٹر کے نیچے دکھائے گئے نتیجے کو پڑھیں

The Quadratic Formula

The quadratic formula solves any equation of the form ax² + bx + c = 0: x = (−b ± sqrt(b² − 4ac)) / 2a. It always works, regardless of whether the equation factors neatly.

The ± means there are generally two solutions (roots). Example: 2x² + 5x − 3 = 0. Here a=2, b=5, c=−3. Discriminant = 25 + 24 = 49. x = (−5 ± 7) / 4, giving x = 0.5 and x = −3.

The Discriminant: How Many Solutions?

The discriminant (b² − 4ac) tells you everything about the solutions:

Positive: Two distinct real roots (parabola crosses x-axis twice)
Zero: One repeated real root (parabola touches x-axis at vertex)
Negative: No real roots — two complex/imaginary roots (parabola does not cross x-axis)

Checking the discriminant first saves time: if it is negative, you know immediately there are no real solutions.

Alternative Methods for Solving Quadratics

Factoring: Fast when it works, but not all quadratics factor neatly. x² + 5x + 6 = (x+2)(x+3) = 0, so x = −2 or x = −3. Completing the square: Always works, useful for deriving vertex form. Graphing: Plot y = ax² + bx + c and find where it crosses the x-axis. The quadratic formula is the most universal method.

Frequently Asked Questions

What if a = 0?

If a = 0, it is not a quadratic equation — it becomes a linear equation (bx + c = 0) with one solution: x = −c/b. The quadratic formula requires a to be nonzero.

What are complex/imaginary roots?

When the discriminant is negative, the square root of a negative number involves i (the imaginary unit, where i² = −1). The roots are x = (−b ± i×sqrt(|discriminant|)) / 2a. These have real-world applications in electrical engineering, signal processing, and quantum mechanics.

How do I find the vertex of a parabola?

The vertex x-coordinate is x = −b/(2a). Plug that back into the equation to get the y-coordinate. The vertex is the minimum point if a > 0 (opens upward) or maximum if a < 0 (opens downward).

💡 کیا آپ جانتے ہیں؟

The quadratic formula was first described in full generality by the Indian mathematician Brahmagupta around 628 AD.
The word "quadratic" comes from the Latin "quadratum" (square) — referring to the x² term that defines these equations.
The discriminant (b²−4ac) tells you everything: if positive, two real roots; if zero, one real root; if negative, two complex (imaginary) roots.

آخری اپ ڈیٹ: March 2026