Introduction to Quadratic Equations

TL;DR

Quadratic equations are special equations where the highest power of the variable is two. They typically look like ax² + bx + c = 0 and have up to two solutions. Solving them helps you find where a parabola crosses the x-axis.

1. The Mental Model

Think of a quadratic equation as describing a curve that looks like a "U" or an upside-down "U" when graphed. Solving the equation means finding the exact points where this curve hits the flat ground (the x-axis).

2. The Core Material

A quadratic equation is a polynomial equation of the second degree. This means the highest power of the unknown variable (usually x) is 2. The general form is:

ax² + bx + c = 0

Here, a, b, and c are real numbers, and a cannot be zero. If a were zero, it would no longer be a quadratic equation, but a linear one (bx + c = 0).

• ax² is the quadratic term.
• bx is the linear term.
• c is the constant term.

Understanding the Solutions (Roots)

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The solutions to a quadratic equation are also called its roots. These are the values of x that make the equation true. Graphically, the roots are the x-intercepts of the parabola that the quadratic equation represents. A quadratic equation can have:

• Two distinct real roots: The parabola crosses the x-axis at two different points.
• One real root (a repeated root): The parabola touches the x-axis at exactly one point (its vertex is on the x-axis).
• No real roots (two complex roots): The parabola does not intersect the x-axis at all.

Methods for Solving Quadratic Equations

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There are several ways to solve quadratic equations:

1. Factoring: If the quadratic expression can be factored into two linear expressions, you can set each factor to zero and solve for x. This is often the quickest method when applicable.
2. Using the Quadratic Formula: This is a universal method that always works, regardless of whether the equation is factorable. The formula is:
   x = [-b ± sqrt(b² - 4ac)] / 2a
   The term b² - 4ac is called the discriminant, which tells you about the nature of the roots.
3. Completing the Square: This method transforms the equation into a perfect square trinomial, making it easy to solve. It's often used to derive the quadratic formula itself.
4. Graphing: By plotting the parabola y = ax² + bx + c, the x-intercepts will be your solutions. This gives a visual understanding but might not yield precise answers.

Here's a flowchart showing the decision process for choosing a solution method:

graph TD
    A["Start Solving ax² + bx + c = 0"] --> B{"Is it easy to factor?"};
    B -- Yes --> C["Solve by Factoring"];
    B -- No --> D{"Is the 'b' term missing (bx=0)?"};
    D -- Yes --> E["Isolate x² and take square root"];
    D -- No --> F{"Do you need exact roots for any quadratic?"};
    F -- Yes --> G["Use Quadratic Formula: x = [-b ± sqrt(b² - 4ac)] / 2a"];
    F -- No --> H["Use Completing the Square (or check graph)"];

3. Worked Example

Let's solve the quadratic equation 2x² + 5x - 3 = 0 using the quadratic formula.

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

The quadratic formula is:
x = [-b ± sqrt(b² - 4ac)] / 2a

Substitute the values:
x = [-5 ± sqrt(5² - 4 * 2 * -3)] / (2 * 2)
x = [-5 ± sqrt(25 - (-24))] / 4
x = [-5 ± sqrt(25 + 24)] / 4
x = [-5 ± sqrt(49)] / 4
x = [-5 ± 7] / 4

Now we find the two possible solutions:

x1 = (-5 + 7) / 4 = 2 / 4 = 1/2
x2 = (-5 - 7) / 4 = -12 / 4 = -3

So, the solutions (roots) for the equation 2x² + 5x - 3 = 0 are x = 1/2 and x = -3.

4. Key Takeaways

• A quadratic equation has the form ax² + bx + c = 0, where a cannot be zero.
• The solutions to a quadratic equation are called roots and represent where its graph crosses the x-axis.
• Quadratic equations can have two, one, or no real solutions.
• Factoring is a quick method if the equation is easily factorable.
• The quadratic formula (x = [-b ± sqrt(b² - 4ac)] / 2a) always works for finding roots.
• The discriminant (b² - 4ac) tells you if there are two, one, or no real roots.
• Completing the square is another powerful algebraic method for solving.

Common Mistakes to Avoid:
- Forgetting that a cannot be zero in a quadratic equation.
- Incorrectly applying the order of operations (PEMDAS/BODMAS) when using the quadratic formula, especially with negative numbers.
- Not looking for two possible solutions when using the ± in the quadratic formula, unless the discriminant is zero.
- Trying to factor every equation; some aren't factorable over integers, so the quadratic formula is necessary.

5. Now Try It

Solve the quadratic equation x² - x - 12 = 0 using both factoring and the quadratic formula.

What to do:
1. Identify a, b, and c for the given equation.
2. First, try to factor the quadratic expression into two binomials. Set each binomial equal to zero to find the roots.
3. Next, use the quadratic formula with your identified a, b, and c values to find the roots.
4. Compare your answers from both methods.

What success looks like: You should get the same two solutions for x using both methods.