October 2, 2025

What Is The Quadratic Formula

Q: What is the part under the square root (b² - 4ac) called?

That part of the formula is called the discriminant. It determines the number and type of solutions: if it's positive, there are two distinct real solutions; if it's zero, there is exactly one real solution; and if it's negative, there are two complex solutions.

Q: When should I use the quadratic formula instead of factoring?

Use the quadratic formula when factoring the equation is not obvious, seems too complex, or doesn't work. The formula is a universal method that works for all quadratic equations, whereas factoring is a shortcut that only works for some.

Q: Why is there a plus-minus sign (±) in the formula?

The plus-minus sign (±) signifies that there are typically two possible solutions for a quadratic equation. You must perform the calculation twice: once using addition (+) and once using subtraction (-) to find both roots.

Q: What happens if the 'a' coefficient is zero?

If 'a' is zero, the equation is not quadratic; it becomes a linear equation (bx + c = 0). The quadratic formula cannot be used because it would result in division by zero. You would solve the simpler linear equation instead.

Learn what the quadratic formula is, how to use it to solve quadratic equations, and see a step-by-step example. A key tool in algebra for finding unknown roots.

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What Is the Quadratic Formula?

The quadratic formula is a mathematical formula used to find the solutions (or roots) of a quadratic equation, which is a second-degree polynomial equation of the form ax² + bx + c = 0. The formula itself is expressed as: x = [-b ± √(b² - 4ac)] / 2a.

Section 2: Understanding the Components

In the formula, 'x' represents the unknown variable you are solving for. The letters 'a', 'b', and 'c' are the numerical coefficients from the quadratic equation: 'a' is the coefficient of the x² term, 'b' is the coefficient of the x term, and 'c' is the constant term. You must identify these three values correctly from the equation before using the formula.

Section 3: A Practical Example

Let's solve the equation 2x² + 5x - 3 = 0. Here, a=2, b=5, and c=-3. Plugging these values into the formula gives: x = [-5 ± √(5² - 4(2)(-3))] / 2(2). This simplifies to x = [-5 ± √(25 + 24)] / 4, which becomes x = [-5 ± √49] / 4. The two solutions are x = (-5 + 7)/4 = 0.5 and x = (-5 - 7)/4 = -3.

Section 4: Why Is the Formula Important?

The quadratic formula is a fundamental tool in algebra because it provides a reliable method to solve any quadratic equation, especially those that are difficult or impossible to factor by hand. It guarantees a path to finding the roots, which are critical in fields like physics for analyzing projectile motion, engineering for design optimization, and economics for modeling profit.

Frequently Asked Questions

What is the part under the square root (b² - 4ac) called?

When should I use the quadratic formula instead of factoring?

Why is there a plus-minus sign (±) in the formula?

What happens if the 'a' coefficient is zero?