The real definition of solving a quadratic equation using the quadratic formula is the process of finding the roots (or solutions) of a quadratic equation of the form:

$ax^2 + bx + c = 0$

where $a$, $b$, and $c$ are constants, and $a \neq 0$. The quadratic formula is expressed as:

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

Steps to solve a quadratic equation using the quadratic formula:

1. Identify the coefficients $a$, $b$, and $c$ from the quadratic equation $ax^2 + bx + c = 0$.
2. Calculate the discriminant, $D = b^2 - 4ac$. The discriminant determines the nature of the roots:
   • If $D > 0$, there are two real and distinct roots.
   • If $D = 0$, there is exactly one real root (a repeated root).
   • If $D < 0$, there are no real roots (the solutions are complex).
3. Substitute the values of $a$, $b$, and the discriminant $D$ into the quadratic formula.
4. Simplify the expression to find the values of $x$, which are the solutions to the equation.

Example:

Given the quadratic equation:

$2x^2 - 4x - 3 = 0$

1. Identify $a = 2$, $b = -4$, and $c = -3$.
2. Compute the discriminant:

$D = (-4)^2 - 4(2)(-3) = 16 + 24 = 40$

3. Apply the quadratic formula:

$x = \frac{-(-4) \pm \sqrt{40}}{2(2)} = \frac{4 \pm \sqrt{40}}{4} = \frac{4 \pm 2\sqrt{10}}{4}$

4. Simplify the expression:

$x = 1 \pm \frac{\sqrt{10}}{2}$

Thus, the solutions are:

$x = 1 + \frac{\sqrt{10}}{2} \quad \text{or} \quad x = 1 - \frac{\sqrt{10}}{2}$

Would you like further details or explanations?

Related Questions:

1. What is the significance of the discriminant in the quadratic formula?
2. How can complex solutions arise from the quadratic formula?
3. How does the quadratic formula relate to completing the square?
4. Are there alternative methods to solve a quadratic equation apart from the quadratic formula?
5. How can you apply the quadratic formula to real-life problems?

Tip: Always check the discriminant before simplifying, as it tells you whether your roots will be real or complex!