The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form:

$ax^2 + bx + c = 0$

where $a$, $b$, and $c$ are constants, and $x$ represents the unknown variable. The quadratic formula provides a solution to this equation by calculating the value(s) of $x$.

The Quadratic Formula

The formula is given by:

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

Steps to Use the Quadratic Formula

1. Identify the coefficients $a$, $b$, and $c$ from your quadratic equation. Ensure the equation is in the standard form $ax^2 + bx + c = 0$.

2. Calculate the discriminant: This is the expression under the square root in the formula, $b^2 - 4ac$. The discriminant determines the nature of the roots:

   • If the discriminant is positive, there are two distinct real solutions.
   • If the discriminant is zero, there is one real solution (a repeated root).
   • If the discriminant is negative, there are two complex solutions.

3. Apply the quadratic formula: Substitute the values of $a$, $b$, and $c$ into the quadratic formula. Perform the calculations to find the values of $x$.

4. Simplify the results: Compute the square root, perform the addition or subtraction, and divide by $2a$ to obtain the final solution(s).

Example Problem

Let's solve the quadratic equation:

$2x^2 - 4x - 6 = 0$

1. Identify the coefficients: $a = 2$, $b = -4$, $c = -6$.

2. Calculate the discriminant:

$\text{Discriminant} = (-4)^2 - 4(2)(-6) = 16 + 48 = 64$

3. Apply the quadratic formula:

$x = \frac{-(-4) \pm \sqrt{64}}{2(2)} = \frac{4 \pm 8}{4}$

4. Simplify the results:

• For the positive root: $x = \frac{4 + 8}{4} = \frac{12}{4} = 3$
• For the negative root: $x = \frac{4 - 8}{4} = \frac{-4}{4} = -1$

So, the solutions are $x = 3$ and $x = -1$.

Would you like more details or have any questions?

Follow-Up Questions:

1. How do you determine the discriminant and what does it signify?
2. What are the possible types of roots based on the discriminant's value?
3. How do you handle quadratic equations where $a = 1$?
4. What is the importance of the $\pm$ symbol in the quadratic formula?
5. How would you solve a quadratic equation with a negative discriminant?
6. Can the quadratic formula be used for equations that aren’t quadratic?
7. How does the quadratic formula compare to factoring methods?
8. What are common errors to avoid when using the quadratic formula?

Tip:

Always double-check your calculations, especially when dealing with the discriminant, as it directly affects the number and type of solutions.