How do you solve quadratic equations using the quadratic formula?

How do you solve quadratic equations using the formula?

The quadratic formula is a universal method used to find the solutions (also called roots or $x$-intercepts) of any quadratic equation, regardless of whether it can be easily factored.

The Quadratic Formula

For any quadratic equation in standard form:

$$ax^2 + bx + c = 0$$

Where $a$, $b$, and $c$ are constants and $a \neq 0$, the solutions for $x$ are given by the formula:

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

Step-by-Step Guide

1. Put the equation in standard form: Rearrange your equation so that all terms are on one side equal to zero ($ax^2 + bx + c = 0$).
2. Identify $a$, $b$, and $c$: Determine the coefficients of $x^2$, $x$, and the constant term.
3. Substitute the values: Plug $a$, $b$, and $c$ into the quadratic formula.
4. Simplify the expression:

• Evaluate the discriminant ($b^2 - 4ac$).
• Take the square root.
• Perform the $\pm$ operation to get two separate equations (one for addition, one for subtraction).

1. Write the final solutions: Simplify the resulting fractions to find your final values for $x$.

Example

Solve $x^2 - 5x + 6 = 0$.

• Here, $a = 1$, $b = -5$, and $c = 6$.
• Plug into the formula: $x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)}$
• Simplify: $x = \frac{5 \pm \sqrt{25 - 24}}{2}$
• Simplify further: $x = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2}$
• Calculate the two roots:
• $x = \frac{5 + 1}{2} = \frac{6}{2} = 3$
• $x = \frac{5 - 1}{2} = \frac{4}{2} = 2$

The solutions are $x = 3$ and $x = 2$.