The formula in the image you provided is the quadratic formula:

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

You use this formula to solve quadratic equations of the form:

$ax^2 + bx + c = 0$

Here's when and why you use it:

When to Use:

1. Standard Form: The quadratic equation is in the form $ax^2 + bx + c = 0$.
2. Exact Solutions: When you need precise solutions (exact roots) for the equation, especially if factoring is difficult or impossible.
3. Non-Factorable Quadratics: If the quadratic does not factor neatly into integers or simple fractions.
4. Complex Roots: When the discriminant ($b^2 - 4ac$) is negative, and the roots are complex numbers.
5. General Case: It works for any quadratic equation, regardless of its specific form.

How to Use:

• Identify $a$, $b$, and $c$ from the quadratic equation.
• Compute the discriminant ($b^2 - 4ac$).
• Substitute $a$, $b$, and the discriminant into the formula.
• Simplify to find the two possible values for $x$ (using $+$ and $-$ for the $\pm$ part).

Would you like a detailed step-by-step example of applying the formula?

Here are 5 related questions to expand on this:

1. What does the discriminant ($b^2 - 4ac$) tell you about the nature of the roots?
2. Can you use the quadratic formula for equations that are not in standard form?
3. How does the quadratic formula relate to completing the square?
4. Are there cases where factoring is more efficient than using the quadratic formula?
5. How do you apply the formula to solve quadratics with complex roots?

Tip:

Always simplify the quadratic equation first (if possible) to reduce calculation errors when applying the formula.