Solving Quadratic Equations | Math Notes

Quadratic equations model parabolas and appear throughout mathematics, physics, and engineering. Understanding how to solve them and interpret their solutions is fundamental to algebra and beyond.

1. What is a Quadratic Equation?

A quadratic equation is an equation where the highest power of the variable is 2.

Standard Form:

$$ax^2 + bx + c = 0$$

$a$, $b$, $c$

are numbers (constants)

$a \neq 0$

$a$ cannot be zero

Solutions

values of $x$ that make it true

💡 What does "finding the solution" mean?

Finding the solution means: What value(s) of $x$ make the equation equal to 0?

A quadratic equation can have:

✓ Two real solutions - the parabola crosses the x-axis twice
✓ One real solution - the parabola touches the x-axis once
✓ No real solutions - the parabola doesn't cross the x-axis

2. Solution Methods

Depending on the equation, we can solve it by:

1. Factoring

When the equation factors nicely

$(x + a)(x + b) = 0$

2. Quadratic Formula

Works for all equations

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

3. Special Patterns

Perfect squares

$(x + a)^2 = 0$

The Quadratic Formula

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

This formula works for any quadratic equation in standard form

3. The Discriminant

The discriminant is the part under the square root in the quadratic formula:

$$\Delta = b^2 - 4ac$$

The discriminant tells us how many real solutions exist

Discriminant Value	What Happens	Number of Real Solutions
Positive ($> 0$)	Parabola crosses x-axis	2 solutions
Zero ($= 0$)	Parabola touches x-axis	1 solution
Negative ($< 0$)	Parabola doesn't cross	0 real solutions

4. Worked Examples

Example A: $x^2 + 0.5x - 14 = 0$

Why we use the quadratic formula

This equation does not factor nicely, so the quadratic formula is the best choice.

Given:

$a = 1$
$b = 0.5$
$c = -14$

Step 1: Find the discriminant

$$b^2 - 4ac = (0.5)^2 - 4(1)(-14) = 0.25 + 56 = 56.25$$

Step 2: Take the square root

$$\sqrt{56.25} = 7.5$$

Step 3: Plug into the formula

$$x = \frac{-0.5 \pm 7.5}{2}$$

Solutions:

$$x = 3.5 \quad \text{and} \quad x = -4$$

Why there are two solutions

Because the discriminant was positive, the graph crosses the x-axis in two places.

Example B: $x^2 + 12x + 36 = 0$

Why factoring works

This equation is a perfect square.

Factor:

$$x^2 + 12x + 36 = (x + 6)^2$$

Solve:

$$(x + 6)^2 = 0$$ $$x + 6 = 0$$ $$x = -6$$

Solution:

$$x = -6$$

Why there is only one solution

The parabola just touches the x-axis at one point. This is called a double root.

Example C: $x^2 - 3x + 8 = 0$

Check the discriminant:

$$(-3)^2 - 4(1)(8) = 9 - 32 = -23$$

What this means

• The discriminant is negative
• You cannot take the square root of a negative number in real numbers

Conclusion:

There are no real solutions

Note: If complex numbers are allowed, the solutions exist but include $i$, the imaginary unit.

Example D: $x^2 + 4 = 0$

Solve:

$$x^2 = -4$$

Why there are no real solutions

A square number is never negative, so this equation has no real solution.

Conclusion:

No real solutions

Optional: Complex solutions would be $x = \pm 2i$

5. Big Picture Summary

Discriminant Value	What Happens	Number of Real Solutions
Positive	Crosses x-axis	2 solutions
Zero	Touches x-axis	1 solution
Negative	No crossing	0 real solutions

6. What You Should Remember

• Quadratic equations model parabolas

• Different equations require different solving methods

• The discriminant tells you how many solutions exist

• Always choose the simplest method first

💡 Pro Tip

Before using the quadratic formula, always check if the equation factors easily or if it's a perfect square. This can save you time and reduce errors!