Determining Real Number Solutions Of Quadratic Functions Using The Discriminant

In this article, we will explore how to determine the number of real-number solutions a quadratic function has using the discriminant. We will use the quadratic function $y = 5x^2 + 5x + 21$ as an example and walk through the process step by step. Understanding the discriminant is crucial for solving quadratic equations and analyzing their nature.

Understanding Quadratic Functions and Their Solutions

Before diving into the discriminant, let's briefly discuss quadratic functions and their solutions. A quadratic function is a polynomial function of the form:

$f(x) = ax^2 + bx + c$

where a, b, and c are constants, and a is not equal to 0. The solutions to the quadratic equation $ax^2 + bx + c = 0$ are the values of x that make the equation true. These solutions are also known as the roots or zeros of the quadratic function. Graphically, they represent the points where the parabola (the graph of the quadratic function) intersects the x-axis.

A quadratic equation can have:

• Two distinct real solutions: The parabola intersects the x-axis at two different points.
• One real solution (a repeated root): The parabola touches the x-axis at one point (the vertex).
• No real solutions: The parabola does not intersect the x-axis.

To determine which of these cases applies to a given quadratic equation, we use the discriminant.

The Discriminant: A Key to Unlocking Solutions

The discriminant is a part of the quadratic formula that helps us determine the nature and number of solutions of a quadratic equation. The quadratic formula is given by:

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

The discriminant is the expression under the square root:

$D = b^2 - 4ac$

The value of the discriminant tells us how many real solutions the quadratic equation has:

• If $D > 0$, the equation has two distinct real solutions.
• If $D = 0$, the equation has one real solution (a repeated root).
• If $D < 0$, the equation has no real solutions (the solutions are complex).

The discriminant is a powerful tool because it allows us to quickly determine the number of real solutions without actually solving the quadratic equation. This is particularly useful in many mathematical and real-world applications.

Applying the Discriminant to Our Example

Now, let's apply the discriminant to the given quadratic function:

$y = 5x^2 + 5x + 21$

Here, we have:

• $a = 5$
• $b = 5$
• $c = 21$

We can now calculate the discriminant:

$D = b^2 - 4ac = 5^2 - 4(5)(21) = 25 - 420 = -395$

Since the discriminant $D = -395$ is less than 0, we can conclude that the quadratic function has no real solutions.

Step-by-Step Calculation

1. Identify the coefficients: In the quadratic function $y = 5x^2 + 5x + 21$, we identify $a = 5$, $b = 5$, and $c = 21$.
2. Write down the discriminant formula: The discriminant is given by $D = b^2 - 4ac$.
3. Substitute the values: Substitute the values of $a$, $b$, and $c$ into the formula: $D = 5^2 - 4(5)(21)$.
4. Calculate the discriminant:
   • $5^2 = 25$
   • $4(5)(21) = 4(105) = 420$
   • $D = 25 - 420 = -395$
5. Interpret the result: Since $D = -395 < 0$, the quadratic function has no real solutions.

Visualizing the Solution

To better understand why there are no real solutions, we can think about the graph of the quadratic function. Since $a = 5$ is positive, the parabola opens upwards. The negative discriminant indicates that the vertex of the parabola is above the x-axis, and the parabola never intersects the x-axis. Therefore, there are no real values of x for which $y = 0$.

Understanding the Implications of the Discriminant's Value

As we've seen, the discriminant is a crucial indicator of the nature of the solutions to a quadratic equation. It's essential to understand what each possible value of the discriminant implies:

Discriminant Greater Than Zero ($D > 0$)

When the discriminant is greater than zero, the quadratic equation has two distinct real solutions. This means the parabola intersects the x-axis at two different points. These points represent the two real roots of the equation. In practical terms, this could represent scenarios where there are two different times when a projectile reaches a certain height, or two different prices at which a business breaks even.

Mathematically, a positive discriminant tells us that the square root in the quadratic formula will yield a real number, leading to two distinct values for x. The formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ clearly shows the $\pm$ sign giving us two different roots.

Discriminant Equal to Zero ($D = 0$)

If the discriminant is equal to zero, the quadratic equation has exactly one real solution, which is often referred to as a repeated root or a double root. Graphically, this means the parabola touches the x-axis at exactly one point, which is the vertex of the parabola. This situation arises when the vertex of the parabola lies on the x-axis.

In practical applications, this could represent a scenario where a system has a critical point or an equilibrium. Mathematically, when $D = 0$, the term under the square root in the quadratic formula is zero, and the formula simplifies to $x = \frac{-b}{2a}$, giving us one unique solution.

Discriminant Less Than Zero ($D < 0$)

When the discriminant is less than zero, the quadratic equation has no real solutions. This means the parabola does not intersect the x-axis at any point. Instead, the solutions are complex numbers, involving the imaginary unit i, where $i = \sqrt{-1}$. Complex solutions indicate that the problem, in the context of real numbers, has no solution. For example, if we are modeling the height of a projectile, a negative discriminant would indicate that the projectile never reaches a certain height.

Mathematically, a negative discriminant means that we are trying to take the square root of a negative number, which yields an imaginary number. This results in complex solutions that do not have a graphical representation on the real number plane.

Real-World Applications of the Discriminant

The discriminant is not just a theoretical concept; it has numerous applications in various fields:

Physics

In physics, the discriminant is used to analyze projectile motion. For example, if we have an equation that models the height of a ball thrown into the air, the discriminant can tell us whether the ball will reach a certain height. If the discriminant is negative, the ball will not reach that height; if it is zero, the ball will reach that height at one point; and if it is positive, the ball will reach that height at two different times (once on the way up and once on the way down).

Engineering

Engineers use the discriminant in structural analysis and design. When designing bridges or buildings, engineers need to ensure that the structures can withstand certain loads and stresses. The discriminant can help determine whether a structure will reach a critical point or fail under certain conditions. For instance, in electrical engineering, the discriminant can help determine the stability of a circuit.

Economics

In economics, quadratic equations often appear in cost and revenue models. The discriminant can be used to determine the break-even points of a business, where the revenue equals the cost. A positive discriminant would indicate two break-even points, a zero discriminant would indicate one, and a negative discriminant would indicate that the business never breaks even under the given model.

Computer Graphics

In computer graphics and game development, the discriminant is used in collision detection. Determining whether two objects collide often involves solving quadratic equations. The discriminant can help quickly determine whether a collision will occur, which is crucial for realistic simulations.

Conclusion

In summary, the discriminant ($b^2 - 4ac$) is a powerful tool for determining the number of real-number solutions of a quadratic equation. By calculating the discriminant, we can quickly determine whether a quadratic equation has two distinct real solutions, one real solution, or no real solutions. For the given quadratic function $y = 5x^2 + 5x + 21$, the discriminant is -395, which is less than 0, indicating that the function has no real solutions. Understanding the discriminant is crucial for solving quadratic equations and analyzing their behavior in various mathematical and real-world contexts.