What if the Discriminant is 4?

If the discriminant of a quadratic equation is 4, it means the equation has two real and distinct roots. This specific value indicates a clear and predictable outcome for the solutions of the quadratic equation.

Understanding the Discriminant

The discriminant is a crucial part of the quadratic formula, used to solve equations of the form $ax^2 + bx + c = 0$. It is represented by the Greek letter delta ($\Delta$) and calculated as:

$\Delta = b^2 - 4ac$

Where:

• $a$, $b$, and $c$ are the coefficients of the quadratic equation.

The value of the discriminant determines the nature of the roots (solutions) of the quadratic equation. Here's a quick overview:

Implications When the Discriminant is 4

When the discriminant is exactly 4, it falls under the $\Delta > 0$ category, confirming that there are two distinct real solutions.

Here's why:

1. Quadratic Formula: The roots of a quadratic equation are found using the formula:
   $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
2. Substituting the Discriminant: Since the discriminant is $b^2 - 4ac$, we can substitute it into the formula:
   $x = \frac{-b \pm \sqrt{\Delta}}{2a}$
3. Specific Case ($\Delta = 4$): When $\Delta = 4$, the formula becomes:
   $x = \frac{-b \pm \sqrt{4}}{2a}$
   $x = \frac{-b \pm 2}{2a}$
4. Two Distinct Roots: This yields two unique real roots:
   • $x_1 = \frac{-b + 2}{2a}$
   • $x_2 = \frac{-b - 2}{2a}$
     Because we are adding and subtracting a real number (2) from $-b$, these two expressions will always result in two different real numbers, assuming $a \neq 0$.

Graphically, a quadratic equation with a discriminant of 4 means its corresponding parabola will intersect the x-axis at two separate points. These intersection points are the real roots of the equation.

Practical Example

Let's consider a quadratic equation where the discriminant is 4.

Equation: $x^2 - 6x + 7 = 0$

1. Identify Coefficients:

   • $a = 1$
   • $b = -6$
   • $c = 7$

2. Calculate the Discriminant:
   $\Delta = b^2 - 4ac$
   $\Delta = (-6)^2 - 4(1)(7)$
   $\Delta = 36 - 28$
   $\Delta = 8$

   Wait, the discriminant here is 8, not 4. Let's create an example that does result in a discriminant of 4 to demonstrate the concept clearly.

   Let's construct an equation where $\Delta = 4$.
   Suppose $a=1$, $b=4$. We need $b^2 - 4ac = 4$.
   $(4)^2 - 4(1)c = 4$
   $16 - 4c = 4$
   $12 = 4c$
   $c = 3$

   So, the equation is $x^2 + 4x + 3 = 0$.

   Let's verify the discriminant for $x^2 + 4x + 3 = 0$:

   • $a = 1$, $b = 4$, $c = 3$
   • $\Delta = (4)^2 - 4(1)(3) = 16 - 12 = 4$. This works!

3. Solve for the Roots using the Quadratic Formula:
   $x = \frac{-b \pm \sqrt{\Delta}}{2a}$
   $x = \frac{-(4) \pm \sqrt{4}}{2(1)}$
   $x = \frac{-4 \pm 2}{2}$

   Now, find the two distinct roots:

   • $x_1 = \frac{-4 + 2}{2} = \frac{-2}{2} = -1$
   • $x_2 = \frac{-4 - 2}{2} = \frac{-6}{2} = -3$

As expected, the equation $x^2 + 4x + 3 = 0$, with a discriminant of 4, yields two distinct real roots: $-1$ and $-3$.

Why This Matters

Understanding the discriminant's value is fundamental in algebra and pre-calculus:

• Predicting Root Nature: It allows you to determine the type and number of solutions without fully solving the quadratic equation. This is particularly useful in higher-level mathematics.
• Graphing Quadratics: Knowledge of the discriminant helps visualize how the parabola defined by the quadratic equation interacts with the x-axis, confirming if it crosses twice, touches once, or doesn't cross at all.
• Problem Solving: In various applications, knowing the nature of solutions (e.g., whether real solutions exist for a physical problem) can be more important than the exact values themselves.