Mastering the Quadratic Formula: A Comprehensive Guide to Solving Quadratic Equations

quadratic formula

The quadratic formula is a key tool in solving quadratic equations

The quadratic formula is a key tool in solving quadratic equations. A quadratic equation is usually written in the form of ax^2 + bx + c = 0, where a, b, and c are constants.

The quadratic formula allows us to find the solutions (also known as roots) of a quadratic equation. The formula states:

x = (-b ± √(b^2 – 4ac)) / (2a)

Let’s break down the components of the formula:

1. The “±” symbol indicates that there are two possible solutions to a quadratic equation. This is because a quadratic equation can have two real and distinct roots, one real root (if the discriminant is zero), or two complex roots (if the discriminant is negative).

2. The discriminant, which is the expression inside the square root (√), is represented as b^2 – 4ac. This value helps us determine the nature of the solutions. If the discriminant is positive, we have two distinct real solutions. If the discriminant is zero, we have one real solution. If the discriminant is negative, we have two complex solutions.

3. “a”, “b”, and “c” represent the coefficients of the quadratic equation. The coefficient “a” is the coefficient of the x^2 term, “b” is the coefficient of the x term, and “c” is the constant term.

To use the quadratic formula, plug in the values of the coefficients (a, b, and c) into the formula and simplify the expression.

For example, let’s solve the quadratic equation 2x^2 + 5x – 3 = 0 using the quadratic formula.

We identify a = 2, b = 5, and c = -3. Plugging these values into the quadratic formula gives us:

x = (-5 ± √(5^2 – 4(2)(-3))) / (2(2))

Simplifying further, we have:

x = (-5 ± √(25 + 24)) / 4
= (-5 ± √49) / 4
= (-5 ± 7) / 4

This yields two possible solutions:

x = (-5 + 7) / 4 = 2/4 = 1/2
x = (-5 – 7) / 4 = -12/4 = -3

Therefore, the solutions to the quadratic equation 2x^2 + 5x – 3 = 0 are x = 1/2 and x = -3.

Remember to always simplify and check your answers to ensure accuracy. The quadratic formula is a powerful tool for solving a wide range of quadratic equations.

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