Skip to content
Mom's Basement University

Solving Quadratic Equations

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn four reliable methods for solving quadratic equations and when to choose each one.

The Concept

Section titled “The Concept”

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a ≠ 0.

Four main solving methods:

  1. Factoring - fastest when the equation factors nicely into two binomials.
  2. Square Root Method - best when the equation is in the form x² = k or after completing the square.
  3. Completing the Square - works for any quadratic and connects nicely to vertex form.
  4. Quadratic Formula - works for every quadratic equation (most reliable method).

The Quadratic Formula is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The expression b² − 4ac is called the discriminant:

  • Positive → two distinct real solutions
  • Zero → one real solution (repeated root)
  • Negative → two complex (imaginary) solutions

Worked Example

Section titled “Worked Example”

Solve 2x² − 8x + 6 = 0 using different methods.

The graph shows the simplified version x² − 4x + 3 (after dividing by 2). The gold dots where the curve crosses the x-axis are the solutions, x = 1 and x = 3. Solving a quadratic equation means finding these x-intercepts. Let’s verify with all three algebraic methods.

Method 1: Factoring

Divide both sides by 2 first: x² − 4x + 3 = 0. Now find two numbers that multiply to 3 and add to −4. Those are −1 and −3:

(x−1)(x−3)=0(x - 1)(x - 3) = 0

Set each factor equal to zero: x = 1 or x = 3.

Method 2: Quadratic Formula

Using the original equation 2x² − 8x + 6 = 0, we have a = 2, b = −8, c = 6.

First, find the discriminant:

b2−4ac=(−8)2−4(2)(6)=64−48=16b^2 - 4ac = (-8)^2 - 4(2)(6) = 64 - 48 = 16

Since the discriminant is positive (16), we know there are two distinct real solutions. Now plug into the formula:

x=−(−8)±162(2)=8±44x = \frac{-(-8) \pm \sqrt{16}}{2(2)} = \frac{8 \pm 4}{4} x=8+44=3orx=8−44=1x = \frac{8 + 4}{4} = 3 \qquad \text{or} \qquad x = \frac{8 - 4}{4} = 1

Method 3: Completing the Square

Start with x² − 4x + 3 = 0 (after dividing by 2). Move the constant to the right:

x2−4x=−3x^2 - 4x = -3

Take half of −4 (which is −2), square it (which is 4), and add to both sides:

x2−4x+4=−3+4x^2 - 4x + 4 = -3 + 4 (x−2)2=1(x - 2)^2 = 1

Take the square root of both sides:

x−2=±1x - 2 = \pm 1

So x = 2 + 1 = 3 or x = 2 − 1 = 1.

All three methods give the same solutions: x = 1 and x = 3.

Real-World Application

Section titled “Real-World Application”

Quadratic equations model many real situations:

  • Projectile motion: finding when a ball or rocket will hit the ground.
  • Business: maximizing profit or finding break-even points.
  • Area optimization: finding dimensions that maximize area with a fixed perimeter.
  • Physics: calculating time for an object to reach a certain height or speed.

Example: a ball is thrown with height h(t) = −16t² + 32t + 48. Setting h(t) = 0 and solving tells you the times when the ball hits the ground.

Quiz

Section titled “Quiz”
Solve by factoring: $x^2 - 5x - 6 = 0$
What is the discriminant of $3x^2 - 7x + 2 = 0$?
Which method is usually fastest when the quadratic factors easily?
If the discriminant is negative, the equation has:
Solve using the quadratic formula: $2x^2 + 3x - 2 = 0$.