The Quadratic Formula Explained Step-by-Step

What Is a Quadratic Equation?

A quadratic equation is any equation that can be written in the standard form:

ax² + bx + c = 0

Where:

• a, b, and c are numbers (coefficients)
• a ≠ 0 (if a = 0, it's not quadratic — it's linear)
• x is the variable you're solving for

The word "quadratic" comes from "quad" meaning square, because the highest power of x is 2 (x squared).

Examples of quadratic equations:

• x² + 5x + 6 = 0 (a=1, b=5, c=6)
• 2x² - 3x - 2 = 0 (a=2, b=-3, c=-2)
• x² - 4 = 0 (a=1, b=0, c=-4)

The Quadratic Formula

The quadratic formula is a universal tool that solves ANY quadratic equation:

x = (-b ± √(b² - 4ac)) / 2a

This formula always works, regardless of whether the equation can be factored easily or not. It directly gives you the values of x that satisfy the equation.

The ± symbol means you get two answers:

• One using + (addition)
• One using - (subtraction)

Step-by-Step: How to Use It

Follow these steps every time:

Step 1: Write the equation in standard form: ax² + bx + c = 0

Step 2: Identify the values of a, b, and c

Step 3: Plug these values into the quadratic formula

Step 4: Simplify under the square root first (calculate b² - 4ac)

Step 5: Calculate both solutions (one with +, one with -)

Step 6: Simplify your answers

Example 1: Two Real Solutions

Solve: x² + 5x + 6 = 0

Step 1: Already in standard form. ✓

Step 2: a = 1, b = 5, c = 6

Step 3: Plug into the formula: x = (-5 ± √(5² - 4(1)(6))) / 2(1)

Step 4: Simplify under the square root: b² - 4ac = 25 - 24 = 1 x = (-5 ± √1) / 2

Step 5: Calculate both solutions:

• x₁ = (-5 + 1) / 2 = -4/2 = -2
• x₂ = (-5 - 1) / 2 = -6/2 = -3

Answer: x = -2 or x = -3

You can verify: (-2)² + 5(-2) + 6 = 4 - 10 + 6 = 0 ✓

Example 2: One Real Solution

Solve: x² - 6x + 9 = 0

Step 2: a = 1, b = -6, c = 9

Step 3: x = (6 ± √(36 - 36)) / 2

Step 4: b² - 4ac = 36 - 36 = 0

Step 5: x = (6 ± 0) / 2 = 6/2 = 3

Answer: x = 3 (one repeated solution)

When the discriminant equals zero, the parabola just touches the x-axis at one point.

Example 3: No Real Solutions

Solve: x² + 2x + 5 = 0

Step 2: a = 1, b = 2, c = 5

Step 3: x = (-2 ± √(4 - 20)) / 2

Step 4: b² - 4ac = 4 - 20 = -16

Since we can't take the square root of a negative number (in real numbers), there are no real solutions. The parabola doesn't cross the x-axis.

The Discriminant Explained

The expression under the square root, b² - 4ac, is called the discriminant (often written as Δ or D). It tells you how many solutions to expect before you even solve:

| Discriminant (b² - 4ac) | Number of Solutions | What It Means | |---|---|---| | Positive (> 0) | Two different real solutions | Parabola crosses x-axis twice | | Zero (= 0) | One repeated real solution | Parabola touches x-axis once | | Negative (< 0) | No real solutions | Parabola doesn't touch x-axis |

This is a quick way to check your work or predict the answer type.

When to Use the Quadratic Formula

The quadratic formula isn't the only way to solve quadratic equations. Here's when to use each method:

• Factoring: When the equation factors easily (simple numbers)
• Square root method: When the equation is in the form x² = k
• Completing the square: When you need to derive the vertex form
• Quadratic formula: When factoring is difficult or impossible, or when you want a guaranteed method

Pro tip: When in doubt, use the quadratic formula. It always works.

Common Mistakes to Avoid

1. Forgetting the negative sign on b: In -b, the negative applies to the entire value of b, including its sign
2. Not writing in standard form first: Make sure everything is on one side and equals zero
3. Errors in the discriminant: Be careful with signs when computing b² - 4ac
4. Dividing only part of the numerator by 2a: The entire numerator (-b ± √...) is divided by 2a
5. Forgetting the ± symbol: Always calculate both solutions

Summary

The quadratic formula x = (-b ± √(b² - 4ac)) / 2a is a powerful tool that solves any quadratic equation ax² + bx + c = 0. Always identify a, b, and c first, then plug in carefully. Use the discriminant to predict how many solutions you'll get.