What Is a Quadratic Equation?

A quadratic equation is an equation where the variable appears with exponent 2:

\[ax^2 + bx + c = 0 \qquad \text{with } a \neq 0\]

It has at most two real solutions.

Common Forms

Some quadratic equations look simpler than others:

  • Standard form: ( ax^2 + bx + c = 0 )
  • Pure quadratic: ( ax^2 + c = 0 )
  • Spurious quadratic: ( ax^2 + bx = 0 )
  • Perfect square: ( (x + r)^2 = 0 )

Solving Methods

✅ Factoring and the Zero Product Property (ZPP)

If the equation can be factored as:

\[(x - r_1)(x - r_2) = 0\]

Then the solutions are:

\[x = r_1 \quad \text{and} \quad x = r_2\]

✅ Quadratic Formula

For any equation in standard form:

\[ax^2 + bx + c = 0\]

You can use:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

The part under the square root is called the discriminant:

\[\Delta = b^2 - 4ac\]

✅ How Many Solutions?

  • If ( \Delta > 0 ): two real solutions
  • If ( \Delta = 0 ): one real solution (a double root)
  • If ( \Delta < 0 ): no real solution

Solved Examples

Example 1

Solve:

\[x^2 - 9x = 0\]

Factor:

\[x(x - 9) = 0 \Rightarrow x = 0 \quad \text{or} \quad x = 9\]

Example 2

Solve:

\[x^2 + 6x + 9 = 0\]

This is a perfect square:

\[(x + 3)^2 = 0 \Rightarrow x = -3\]

Example 3

Solve:

\[2x^2 + 3x + 1 = 0\]

Use the formula:

\[x = \frac{-3 \pm \sqrt{3^2 - 4\cdot2\cdot1}}{2 \cdot 2} = \frac{-3 \pm \sqrt{1}}{4}\]

So:

\[x = -1 \quad \text{or} \quad x = -\frac{1}{2}\]

Practice Exercises

Try solving the following equations:

  1. ( x^2 - 4 = 0 )
  2. ( x(x - 5) = 0 )
  3. ( x^2 + 8x + 16 = 0 )
  4. ( x^2 + 2x - 3 = 0 )
  5. ( x^2 + x + 1 = 0 )

Suggested Solutions

  1. ( x = -2 ), ( x = 2 )
  2. ( x = 0 ), ( x = 5 )
  3. ( x = -4 )
  4. ( x = 1 ), ( x = -3 )
  5. ❌ No real solutions (discriminant is negative)

📚 Want the Full Lesson?

This is just a quick overview. You can find the complete slide set — with theory, examples, and exercises — in our free explained PDF:

👉 Quadratic Equations – Full Lesson (PDF)

Or explore all available lessons:

👉 High School Explained Slides