What is a Quadratic Equation?

It is any equation that can be written in the form ax² + bx + c = 0. The graph of these equations is always a parabola (a U-shaped curve). The highest exponent (degree) is always 2.

What is the Discriminant?

The Discriminant is the part under the square root symbol (b² - 4ac). It acts as a predictor. If it's Positive, there are 2 real solutions. If it's Zero, there is 1 real solution (the vertex). If it's Negative, there are 2 imaginary (complex) solutions.

Can I just factor it instead?

Sometimes! Factorization (like FOIL in reverse) is faster if the numbers are clean integers (e.g., x² + 5x + 6). But the Quadratic Formula works 100% of the time, even for 'ugly' equations with decimals or irrational roots.

What do the solutions mean?

The solutions (roots or zeros) represent where the parabola crosses the X-axis on a graph. If you threw a ball, the solutions would be the start point and the end point where it hits the ground.

The Weapon of Mass Calculation: The Quadratic Formula

Algebra students dread memorizing it, but it is the skeleton key for math. It solves any quadratic equation, no matter how ugly the numbers are.

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The Song (How to Memorize It)

If you attended high school in the US, you likely learned this to the tune of "Pop Goes the Weasel." Sing it in your head:

X equals negative B,
Plus or minus the square root,
Of B squared minus 4AC,
All over 2A.

The Formula Breakdown

The formula is derived from "Completing the Square" on the standard form equation ax² + bx + c = 0.

x = -b ± √(b² - 4ac)2a

Where:
a = Coefficient of x² (The "Quadratic" term)
b = Coefficient of x (The "Linear" term)
c = Constant term (The intercept)

Note: The equation MUST equal zero. If you have 2x² = 5x + 3, you must move everything to the left side first: 2x² - 5x - 3 = 0.

The Discriminant: The Crystal Ball

The expression inside the square root, b² - 4ac, has a special name: The Discriminant. It tells you the nature of the solutions without solving the whole thing.

Discriminant > 0

Two Real Solutions

Parabola crosses x-axis twice.

Discriminant = 0

One Real Solution

Parabola touches x-axis once (Vertex).

Discriminant < 0

No Real Solutions

Parabola never touches x-axis (Complex roots).

Step-by-Step Example

Let's solve: 2x² - 4x - 6 = 0

a = 2
b = -4 (Don't forget the negative sign!)
c = -6

Step 1: Calculate the Discriminant

b² - 4ac

(-4)² - 4(2)(-6)
= 16 - (-48)
= 16 + 48
= 64

Since 64 is positive and a perfect square (8²), we know we will have two clean, rational answers.

Step 2: Plug into the Formula

x = [-(-4) ± √64] / 2(2)
x = [4 ± 8] / 4

Step 3: Split and Solve

Case 1 (Plus):
x = (4 + 8) / 4 = 12 / 4 = 3

Case 2 (Minus):
x = (4 - 8) / 4 = -4 / 4 = -1

Final Answer: x = 3 and x = -1

Real World Applications: Projectile Motion

Why does this matter? Because gravity is quadratic. Any object thrown in the air follows a parabolic arc described by:

h(t) = -16t² + v₀t + h₀

If you launch a rocket and want to know "When will it hit the ground?", you are asking "When does height = 0?". Solving that quadratic equation gives you the exact time of impact.
(The negative solution represents time before launch, which we ignore in physics).

Conclusion

The Quadratic Formula is reliable. Unlike factoring, which relies on luck and nice numbers, calculating the formula works for every possible parabola in the universe. Master it, and you master Algebra 1.