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Quadratic Equations: Factoring, Formulas, and Parabolas

Learn the core methods for solving quadratic equations: factoring, completing the square, and using the quadratic formula, plus what parabolas look like.

Practice Quadratics

What Are Quadratic Equations?

A quadratic equation has an x^2 term in it (and no higher powers, like x^3). They form beautiful U-shaped curves called parabolas when graphed, and solving them usually means finding where that curve crosses the x-axis. Because of the curve, quadratics often have two solutions.

The standard form is:

ax^2 + bx + c = 0

Method 1: Factoring

Often the fastest way, if the numbers work out nicely. You look for two numbers that multiply to c and add up to b, then rewrite the equation as (x + p)(x + q) = 0.

Method 2: The Quadratic Formula

The heavy-duty tool that works on every single quadratic equation, even when factoring is impossible.

x = [-b ± √(b^2 - 4ac)] / 2a

Worked Examples

Solving by Factoring

Medium

Solve: x^2 + 5x + 6 = 0

Step 1: Set equation to zero
x^2 + 5x + 6 = 0
The equation is already in standard form (ax^2 + bx + c = 0).
Step 2: Find factors
What multiplies to 6 and adds to 5?
2 and 3 fit both conditions.
Step 3: Write as factored expression
(x + 2)(x + 3) = 0
Rewrite the equation using the factors.
Step 4: Solve for x
x = -2 or x = -3
Set each factor to zero to find the solutions.
Final Answer:
x = -2, x = -3

Solving with the Quadratic Formula

Hard

Solve: x^2 - 3x - 4 = 0

Step 1: Identify a, b, c
a = 1, b = -3, c = -4
Read the coefficients from standard form: ax^2 + bx + c = 0.
Step 2: Plug into formula
x = (3 ± √( (-3)^2 - 4(1)(-4) )) / 2(1)
Careful with negative signs when substituting into the formula.
Step 3: Simplify under the root (Discriminant)
x = (3 ± √( 9 + 16 )) / 2
Simplify the expression under the square root.
Step 4: Final Evaluation
x = (3 ± 5) / 2 -> x = 4 or x = -1
Evaluate both the plus and minus scenarios.
Final Answer:
x = 4, x = -1

The Discriminant

The part of the formula under the square root — b^2 - 4ac — is called the discriminant. It tells you instantly how many solutions exist:

  • Positive: Two real solutions (crosses x-axis twice).
  • Zero: One real solution (vertex rests exactly on the x-axis).
  • Negative: No real solutions (parabola never touches the x-axis — leads to complex numbers).

Where this shows up next

Mastering quadratics is critical before taking Precalculus. You'll expand these concepts to graph more complex polynomials, deal with imaginary numbers, and explore conic sections.

Ready to practice? Try the free quadratic equations worksheet: 29 problems on factoring, completing the square and the quadratic formula, with an answer key.

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