Certainly! Let's discuss the detailed, step-by-step reasoning behind the given formula.
The formula [tex]\( x = -b \pm \frac{\sqrt{b^2 - 4ac}}{2a} \)[/tex] represents the solutions to a quadratic equation. A quadratic equation is typically written in the form:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants, and [tex]\( x \)[/tex] is the variable we need to solve for. This formula is known as the quadratic formula.
The steps to derive and understand this formula are as follows:
### Standard Form of a Quadratic Equation
1. Identify the standard form:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
### Using the Quadratic Formula
2. Quadratic formula:
The quadratic formula is:
[tex]\[ x = -b \pm \frac{\sqrt{b^2 - 4ac}}{2a} \][/tex]
### Components of the Formula
3. Discriminant: The term under the square root, [tex]\( b^2 - 4ac \)[/tex], is called the discriminant.
- If the discriminant is positive ([tex]\( b^2 - 4ac > 0 \)[/tex]), there are two distinct real solutions.
- If the discriminant is zero ([tex]\( b^2 - 4ac = 0 \)[/tex]), there is exactly one real solution (also called a repeated root).
- If the discriminant is negative ([tex]\( b^2 - 4ac < 0 \)[/tex]), there are two complex solutions.
4. ± Symbol: The [tex]\( \pm \)[/tex] symbol means that there are generally two solutions to the quadratic equation:
[tex]\[ x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \][/tex]
[tex]\[ x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Completing the Calculation
5. Division by [tex]\( 2a \)[/tex]: Both solutions are divided by [tex]\( 2a \)[/tex] to get the final value of [tex]\( x \)[/tex].
### Example Application
Let's apply the formula to a hypothetical quadratic equation:
[tex]\[ 2x^2 + 4x + 2 = 0 \][/tex]
1. Identify coefficients: [tex]\( a = 2 \)[/tex], [tex]\( b = 4 \)[/tex], [tex]\( c = 2 \)[/tex]
2. Compute the discriminant:
[tex]\[
b^2 - 4ac = 4^2 - 4 \cdot 2 \cdot 2 = 16 - 16 = 0
\][/tex]
3. Plug into the quadratic formula:
[tex]\[
x = \frac{-4 \pm \sqrt{0}}{2 \cdot 2} = \frac{-4 \pm 0}{4} = \frac{-4}{4} = -1
\][/tex]
Here, [tex]\( x = -1 \)[/tex] is the only solution as the discriminant is zero.
In conclusion, [tex]\( x = -b \pm \frac{\sqrt{b^2 - 4ac}}{2a} \)[/tex] provides the solutions to the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]. Whether the equation has two distinct real solutions, one real solution, or two complex solutions depends on the value of the discriminant [tex]\( b^2 - 4ac \)[/tex].
