Quadratic Equations

A quadratic equation is any equation of the form ax^2 + bx + c = 0, where a is not zero. Unlike linear equations that have one solution, quadratics can have two solutions, one solution, or even no real solutions. Let us explore three powerful methods to solve them.

Method 1: Factoring

This is the quickest method when the equation factors nicely.

Worked Example: Solve x^2 - 5x + 6 = 0

Step 1: Factor the left side: (x - 2)(x - 3) = 0

Step 2: Set each factor to zero: x - 2 = 0 or x - 3 = 0

Step 3: Solve: x = 2 or x = 3

Method 2: Completing the Square

Useful when factoring is difficult. We rewrite the equation in the form (x + p)^2 = q.

Worked Example: Solve x^2 + 6x + 2 = 0

Step 1: Move the constant: x^2 + 6x = -2

Step 2: Take half of 6 (which is 3), square it (9), and add to both sides: x^2 + 6x + 9 = 7

Step 3: Write as a perfect square: (x + 3)^2 = 7

Step 4: Take square root: x + 3 = plus or minus root 7

Step 5: Solve: x = -3 + root 7 or x = -3 - root 7

Method 3: The Quadratic Formula

The Quadratic Formula: For ax^2 + bx + c = 0, x = (-b plus or minus root(b^2 - 4ac)) / (2a)

This works for ALL quadratic equations.

Worked Example: Solve 2x^2 + 3x - 5 = 0

Here a = 2, b = 3, c = -5.

b^2 - 4ac = 9 - 4(2)(-5) = 9 + 40 = 49

x = (-3 plus or minus 7) / 4

x = (-3 + 7)/4 = 1 or x = (-3 - 7)/4 = -5/2

x = 1 or x = -2.5

The Discriminant

The expression D = b^2 - 4ac tells you about the nature of the roots:

• If D > 0: Two distinct real roots
• If D = 0: One repeated real root
• If D < 0: No real roots

Nepal Connection: If you throw a ball upward from a rooftop in Kathmandu, its height follows a quadratic equation. The two solutions tell you when the ball is at a certain height -- once going up and once coming down!

Key Takeaways

• Try factoring first -- it is the fastest method
• Use completing the square when factoring fails
• The quadratic formula always works as a last resort
• The discriminant (b^2 - 4ac) tells you how many real solutions exist