Solve Quadratics - When Linear Coefficient is Zero - Algebra-1

1. Fundamental Concepts

Definition: A quadratic equation has the general form where . When the linear coefficient (the coefficient of the x-term, b) is zero, the equation simplifies to .
Core Feature: The equation contains no linear term (bx), leaving only the quadratic term () and the constant term (c). This allows solving by isolating the quadratic term and taking square roots.

Simplified Equation Form: (where , , and c is a constant).
Solution Principle: Rearrange the equation to isolate $x^2$ , then take the square root of both sides. The steps are:
1. (by subtracting c from both sides).
2. (by dividing both sides by a).
3. (by taking the square root of both sides; solutions exist only if ).
Nature of Solutions:
- If , there are two distinct real solutions: and .
- If , there is one real solution (a repeated root): .
- If , there are no real solutions (only complex solutions).

Easy: Solve Solution:
1. Isolate : .
2. Take square roots: .
3. Solutions: and .
Medium: Solve Solution:
1. Isolate : .
2. Divide by 2: .
3. Take square roots: .
4. Solutions: and .
Hard: Solve Solution:
1. Isolate : .
2. Divide by : .
3. Take square roots: .
4. Solutions: and .

Identify the Equation Type: Confirm the linear coefficient , so the equation is in the form .
Isolate the Quadratic Term: Rearrange the equation to get by subtracting c from both sides.
Solve for : Divide both sides by a to isolate , resulting in .
Take Square Roots: Compute the square root of both sides, remembering to include both the positive and negative roots ().
Check for Real Solutions: Ensure ; if not, state that there are no real solutions.
Verify (Optional): Substitute the solutions back into the original equation to confirm they are correct.