Section 8.2: The Quadratic Formula – Study Notes

Quadratic Equations and the Quadratic Formula

Introduction to Quadratic Equations

A quadratic equation is any equation that can be written in the standard form:

• Standard Form: $a x^2 + b x + c = 0$, where $a \neq 0$

• Quadratic equations have at most two real solutions.

There are several methods to solve quadratic equations, including factoring, completing the square, and the quadratic formula. The quadratic formula is a universal method that works for any quadratic equation.

The Quadratic Formula

The quadratic formula provides the solutions to any quadratic equation of the form $a x^2 + b x + c = 0$:

• Formula:

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

• Discriminant: The expression under the square root, $b^2 - 4ac$, is called the discriminant. It determines the nature of the solutions:

  • If $b^2 - 4ac > 0$, there are two distinct real solutions.

  • If $b^2 - 4ac = 0$, there is one real solution (a repeated root).

  • If $b^2 - 4ac < 0$, there are two complex (non-real) solutions.

• Applicability: The quadratic formula can solve any quadratic equation, but sometimes factoring or completing the square may be simpler.

Identifying Coefficients

To use the quadratic formula, first identify the coefficients $a$, $b$, and $c$ from the equation $a x^2 + b x + c = 0$.

• Example 1: $2x^2 + 9x + 5 = 0$

  • $a = 2$

  • $b = 9$

  • $c = 5$

• Example 2: $3x^2 - 2x = 7$

  • First, rewrite as $3x^2 - 2x - 7 = 0$

  • $a = 3$

  • $b = -2$

  • $c = -7$

Using the Quadratic Formula: Step-by-Step

1. Write the equation in standard form: $a x^2 + b x + c = 0$.

2. Identify $a$, $b$, and $c$.

3. Substitute $a$, $b$, and $c$ into the quadratic formula.

4. Simplify under the square root (the discriminant).

5. Calculate the two possible values for $x$ using $+$ and $-$.

6. Simplify the results. If necessary, approximate the answers using a calculator.

Examples

• Example 1: Solve $x^2 - 5x - 14 = 0$

  • $a = 1$, $b = -5$, $c = -14$

  • Substitute into the formula:

  • $x = \frac{ -(-5) \pm \sqrt{ (-5)^2 - 4 \cdot 1 \cdot (-14) } }{ 2 \cdot 1 }$

  • $x = \frac{ 5 \pm \sqrt{ 25 + 56 } }{ 2 }$

  • $x = \frac{ 5 \pm \sqrt{ 81 } }{ 2 }$

  • $x = \frac{ 5 \pm 9 }{ 2 }$

  • So, $x = 7$ or $x = -2$

• Example 2: Solve $x^2 + 6x - 1 = 4$

  • First, rewrite as $x^2 + 6x - 5 = 0$

  • $a = 1$, $b = 6$, $c = -5$

  • $x = \frac{ -6 \pm \sqrt{ 36 - 4 \cdot 1 \cdot (-5) } }{ 2 \cdot 1 }$

  • $x = \frac{ -6 \pm \sqrt{ 36 + 20 } }{ 2 }$

  • $x = \frac{ -6 \pm \sqrt{ 56 } }{ 2 }$

  • $x = \frac{ -6 \pm 2\sqrt{14} }{ 2 }$

  • $x = -3 \pm \sqrt{14}$

Interpreting Expressions with the Quadratic Formula

Expressions of the form $a \pm \sqrt{b} / c$ represent two numbers, one for the $+$ and one for the $-$ sign.

• Example: $\frac{ -1 \pm \sqrt{25} }{ 4 }$

  • $\sqrt{25} = 5$

  • So, $\frac{ -1 + 5 }{ 4 } = 1$ and $\frac{ -1 - 5 }{ 4 } = -1.5$

• Example: $\frac{ 6 \pm \sqrt{75} }{ 2 }$

  • $\sqrt{75} = 5\sqrt{3}$

  • So, $\frac{ 6 + 5\sqrt{3} }{ 2 }$ and $\frac{ 6 - 5\sqrt{3} }{ 2 }$

Practice Problems

• Solve using the quadratic formula and simplify or approximate as needed:

  • $9x^2 - 6x + 1 = 0$

  • $5x^2 - 4x = 2$

  • $5x^2 - 4x - 2 = 0$

  • $6x(x + 3) + 4 = 3x(x + 1)$

  • $3x^2 = 3x + 5$

• For each, write the equation in standard form, identify $a$, $b$, $c$, and apply the quadratic formula.

Common Mistakes to Avoid

• Forgetting to write the equation in standard form before identifying $a$, $b$, and $c$.

• Incorrectly substituting negative values for $b$ or $c$.

• Not simplifying the square root or the final answer completely.

• Forgetting to divide both terms in the numerator by $2a$.

Summary Table: Quadratic Formula Steps

Step	Description
1	Write the equation in standard form: $a x^2 + b x + c = 0$
2	Identify $a$, $b$, and $c$
3	Substitute into the quadratic formula: $x = \frac{ -b \pm \sqrt{ b^2 - 4 a c } }{ 2a }$
4	Simplify the discriminant ($b^2 - 4ac$)
5	Calculate both solutions (using $+$ and $-$)
6	Simplify and, if needed, approximate the answers

Additional Practice

• Solve and approximate to three decimal places if necessary:

  • $3x^2 + 2x - 8 = 0$

  • $u^2 + 4u = 1$

  • $x^2 + 1 = 9x$

  • $(x + 3)(x + 1) = 5$

Key Takeaways

• The quadratic formula is a powerful tool for solving any quadratic equation.

• Always check your work for arithmetic errors and ensure the equation is in standard form before applying the formula.

• Practice is essential for mastering the quadratic formula and recognizing when it is the most efficient method.