Solve Quadratics - When All Three Coefficients are Non-zero - Algebra-1

1. Fundamental Concepts

Definition: A quadratic equation has the general form , where a, b, and c are constants. When all three coefficients are non-zero (, , ), the equation cannot be simplified by removing the linear term or constant term, requiring more systematic methods to solve.
Core Feature: The equation contains all three terms: the quadratic term (), the linear term (bx), and the constant term (c). Solving such equations typically involves factoring (if possible), completing the square, or using the quadratic formula.

Equation Form: with , , .
Solution Principles:
- Factoring: Rewrite the quadratic expression as a product of two binomials, i.e., , then set each factor equal to 0 and solve for x. This works when the quadratic is factorable over integers.
- Completing the Square: Transform the equation into the form by rearranging terms and adding a constant to both sides, then solve using square roots.
- Quadratic Formula: For any quadratic equation, solutions can be found using $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where the discriminant $D = b^2 - 4ac$ determines the nature of the roots:
  - If : Two distinct real roots.
  - If : One real root (a repeated root).
  - If : No real roots (complex roots).

Easy (Factorable): Solve Solution:
1. Factor the quadratic: .
2. Set each factor to 0: or .
3. Solutions: and .
Medium (Factorable with a non-1 leading coefficient): Solve Solution:
1. Factor the quadratic: .
2. Set each factor to 0: or .
3. Solutions: and .
Hard (Using the quadratic formula): Solve Solution:
1. Identify coefficients: , , .
2. Calculate the discriminant: .
3. Apply the quadratic formula:.
4. Solutions: and .

Identify the Equation Type: Confirm , , in .
Check for Factorability:
- Look for two numbers that multiply to and add to b (for factoring by grouping).
- If factorable, factor into binomials and solve.
Use Completing the Square if factoring is difficult:
- Divide by a to make the leading coefficient 1.
- Move the constant term to the right side.
- Add to both sides to complete the square.
- Take square roots and solve for x.
Apply the Quadratic Formula as a universal method:
- Substitute a, b, c into .
- Simplify the result and check the discriminant to describe the roots.
Verify Solutions: Substitute roots back into the original equation to ensure they satisfy it.