The Quadratic Formula | How to Solve Quadratic Equations and Understand the Nature of Solutions

quadratic formula

The quadratic formula is an algebraic formula used to solve quadratic equations

The quadratic formula is an algebraic formula used to solve quadratic equations. A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x represents the unknown variable.

The quadratic formula is derived from completing the square. To find the values of x that satisfy the equation, we can use the quadratic formula:

x = (-b ± √(b^2 – 4ac)) / (2a)

In the formula, ± represents the two possible solutions of x, √ represents the square root operation, and b^2 – 4ac is called the discriminant. The discriminant determines the nature of the solutions:

1. If the discriminant is positive (b^2 – 4ac > 0), the equation has two distinct real solutions.
2. If the discriminant is zero (b^2 – 4ac = 0), the equation has one real solution (also known as a repeated or double root).
3. If the discriminant is negative (b^2 – 4ac < 0), the equation has no real solutions, but it has two complex solutions (conjugate pairs) involving the imaginary unit "i," such as x = (5 + 3i) and x = (5 - 3i). To solve a quadratic equation using the quadratic formula, simply substitute the values of a, b, and c into the formula and calculate the solutions for x.

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