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Factoring quadratics involves rewriting the quadratic equation in a way that it can be expressed as a product of two binomials. This method helps you easily find the roots of the equation. Let's look at the example given: \[ x^{2} - x - 20 = 0 \]
**Step-by-Step Factoring Process:**
1. Ensure the equation is in standard form \[ ax^{2} + bx + c = 0 \] Here, it's already in standard form: \[ x^{2} - x - 20 = 0 \]
2. Identify the coefficients: In this case, \( a = 1 \), \( b = -1 \), and \( c = -20 \).
3. Find two numbers that multiply to the constant term \( c \) (-20) and add up to \( b \) (-1). These numbers will be 4 and -5, since: \[ 4 \times (-5) = -20 \] \[ 4 + (-5) = -1 \]
4. Rewrite the equation by splitting the middle term -x using these numbers: \[ x^{2} + 4x - 5x - 20 = 0 \]
5. Factor by grouping: \[ x(x + 4) - 5(x + 4) = 0 \]
6. Factor out the common binomial: \[ (x + 4)(x - 5) = 0 \]
**Result:** Now you have the equation in a factored form, which you can solve to find the roots.

The quadratic formula is a powerful tool to solve any quadratic equation, especially when factoring is complex. The formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here’s how to use it step-by-step:
1. Identify the coefficients from the standard form \( ax^{2} + bx + c = 0 \).
2. Substitute the values of \( a \), \( b \), and \( c \) into the formula.
3. Calculate the discriminant \( b^2 - 4ac \).
4. Determine the roots using the \( \pm \) symbol to solve for the two possible values of \( x \).
**Example:** For the equation \( x^{2} - x - 20 = 0 \):
- Here, \( a = 1 \), \( b = -1 \), \( c = -20 \).
- Substitute these into the formula: \[ x = \frac{-(-1) \pm \sqrt{(-1)^{2} - 4(1)(-20)}}{2(1)} \]
- Calculate the discriminant: \[ (-1)^{2} - 4(1)(-20) = 1 + 80 = 81 \]
- Solve for \( x \): \[ x = \frac{1 \pm \sqrt{81}}{2} = \frac{1 \pm 9}{2} \]
- This gives: \[ x = \frac{1 + 9}{2} = 5 \] and \[ x = \frac{1 - 9}{2} = -4 \]
This formula always works, even when factoring is difficult!

The roots of a quadratic equation are the values of \( x \) that make the equation true. These roots are also known as solutions or zeros of the equation.
**Ways to Find Roots:** There are various methods to find the roots:

• **Factoring**: As shown in our example, factoring the quadratic equation and then solving for \( x \) gives us the roots.
• **Quadratic Formula**: Using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we can find the exact roots.
• **Graphing**: By plotting the quadratic equation on a graph, the points where the graph intersects the \( x \)-axis are the roots.
• **Completing the Square**: This involves manipulating the equation into a perfect square trinomial to solve for \( x \).

**Interpreting Roots:**
- **Real and Distinct**: If the discriminant \( b^2 - 4ac \) is positive, the equation has two real and distinct roots.
- **Real and Repeated**: If the discriminant is zero, the equation has one real root, repeated twice.
- **Complex:** If the discriminant is negative, the equation has two complex roots.
**Example:** For \( x^{2} - x - 20 = 0 \), the roots found are \( x = 5 \) and \( x = -4 \), both real and distinct because the discriminant (81) is positive.
Understanding the nature of roots helps in predicting the behavior of the quadratic equation.