October 12, 2025

How To Solve Quadratic Equation Quadratic Formula

Q: What does the 'discriminant' tell you?

The discriminant is the part of the quadratic formula under the square root sign, `b^2 - 4ac`. It indicates the nature of the roots: if > 0, there are two distinct real roots; if = 0, there is exactly one real root (a double root); if < 0, there are two distinct complex roots.

Q: Can the quadratic formula solve all quadratic equations?

Yes, the quadratic formula is a universal method that can solve any quadratic equation, regardless of whether its roots are real (rational or irrational) or complex. It is guaranteed to provide solutions when they exist.

Q: Is factoring always easier than using the quadratic formula?

No, factoring is only easier and quicker for specific quadratic equations where the roots are simple integers or fractions. For most quadratic equations, especially those with irrational or complex roots, factoring is not feasible or significantly more difficult. The quadratic formula is consistently applicable and reliable.

Learn the step-by-step process of solving quadratic equations by applying the quadratic formula, including its components and practical examples.

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Understanding the Quadratic Formula

The quadratic formula is a powerful algebraic tool used to find the roots (or solutions) of any quadratic equation. A quadratic equation is a polynomial equation of the second degree, typically written in the standard form `ax^2 + bx + c = 0`, where `x` is the unknown, and `a`, `b`, and `c` are coefficients with `a ≠ 0`. The formula provides a direct method to calculate the values of `x` that satisfy the equation, even when factoring is not straightforward or possible.

Steps to Apply the Quadratic Formula

To solve a quadratic equation using the formula, first ensure the equation is in standard form (`ax^2 + bx + c = 0`). Next, identify the values of the coefficients `a`, `b`, and `c`. These values are then substituted into the quadratic formula: `x = [-b ± sqrt(b^2 - 4ac)] / 2a`. After substitution, carefully simplify the expression, starting with the part under the square root (the discriminant), then evaluating the square root, and finally performing the addition and subtraction for the '±' to find the two possible solutions for `x`.

Practical Example: Solving 2x² + 5x - 3 = 0

Consider the equation `2x² + 5x - 3 = 0`. Here, `a = 2`, `b = 5`, and `c = -3`. Substitute these into the quadratic formula: `x = [-5 ± sqrt(5^2 - 4 * 2 * -3)] / (2 * 2)`. This simplifies to `x = [-5 ± sqrt(25 + 24)] / 4`, which further becomes `x = [-5 ± sqrt(49)] / 4`. Since `sqrt(49) = 7`, the solutions are `x = (-5 + 7) / 4` and `x = (-5 - 7) / 4`. Thus, `x = 2/4 = 1/2` and `x = -12/4 = -3` are the two solutions.

When to Use the Quadratic Formula

The quadratic formula is invaluable because it guarantees a solution for any quadratic equation, regardless of the nature of its roots (real, irrational, or complex). While other methods like factoring or completing the square can be quicker for specific equations, the quadratic formula is a universal method that is always applicable. It is particularly useful when the quadratic expression does not factor easily, or when one needs to determine the exact nature and values of the roots precisely, making it a fundamental tool in algebra, physics, engineering, and various scientific fields.

Frequently Asked Questions

What is a quadratic equation?

What does the 'discriminant' tell you?

Can the quadratic formula solve all quadratic equations?

Is factoring always easier than using the quadratic formula?