Discriminant Of 2x^2+5x-8 A Comprehensive Guide

The discriminant is a crucial concept in mathematics, especially when dealing with quadratic equations and polynomials. It provides valuable information about the nature of the roots (or solutions) of a quadratic equation. In this article, we will delve deep into understanding the discriminant, its formula, and how to interpret its value. We will also apply this knowledge to a specific polynomial, $2x^2 + 5x - 8$, to determine its discriminant and understand what it tells us about the roots of the equation.

What is the Discriminant?

In mathematics, the discriminant is a term that is most commonly associated with quadratic equations. It is a part of the quadratic formula and provides insight into the nature of the roots of the equation. The discriminant helps us determine whether the quadratic equation has two distinct real roots, one real root (a repeated root), or two complex roots. The discriminant is a powerful tool for analyzing quadratic equations without actually solving them. It stems directly from the quadratic formula, which is used to find the solutions (roots) of any quadratic equation in the standard form $ax^2 + bx + c = 0$.

The quadratic formula is given by:

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

Within this formula, the expression under the square root, $b^2 - 4ac$, is what we call the discriminant. The discriminant, often denoted by the Greek letter delta ($\Delta$), is calculated using the coefficients $a$, $b$, and $c$ from the quadratic equation. The value of the discriminant determines the type and number of solutions the quadratic equation has. Understanding the discriminant is fundamental to solving and analyzing quadratic equations in algebra and beyond. It helps in predicting the nature of the solutions without going through the entire process of solving the equation, saving time and effort. Let's dive deeper into how the discriminant works and what it tells us about the roots.

The Formula for the Discriminant

The discriminant formula is a simple yet powerful expression derived from the quadratic formula. It is used to determine the nature and number of solutions (roots) of a quadratic equation. As we discussed, the quadratic formula is given by:

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

The discriminant is the expression under the square root in the quadratic formula:

$\Delta = b^2 - 4ac$

Where:

• $a$ is the coefficient of the $x^2$ term.
• $b$ is the coefficient of the $x$ term.
• $c$ is the constant term.

This formula is straightforward to apply once you've identified the coefficients $a$, $b$, and $c$ from the quadratic equation in the standard form $ax^2 + bx + c = 0$. The discriminant is a single numerical value that can be positive, negative, or zero. Each of these outcomes has a specific meaning regarding the nature of the roots.

• If $\Delta > 0$, the quadratic equation has two distinct real roots. This means there are two different real numbers that satisfy the equation.
• If $\Delta = 0$, the quadratic equation has one real root (a repeated root). This indicates that the parabola representing the quadratic equation touches the x-axis at exactly one point.
• If $\Delta < 0$, the quadratic equation has two complex roots. This means the roots are complex numbers, involving the imaginary unit $i$, and there are no real solutions.

Mastering the discriminant formula is essential for quickly assessing the characteristics of a quadratic equation's solutions. It's a fundamental concept that simplifies the analysis of quadratic equations and their graphical representations.

Interpreting the Value of the Discriminant

The value of the discriminant is a key indicator of the nature of the roots of a quadratic equation. As we've established, the discriminant is calculated using the formula $\Delta = b^2 - 4ac$. The resulting value can be positive, negative, or zero, and each scenario tells us something different about the solutions of the quadratic equation.

Case 1: $\Delta > 0$ (Positive Discriminant)

When the discriminant is positive, i.e., $b^2 - 4ac > 0$, the quadratic equation has two distinct real roots. This means there are two different real numbers that satisfy the equation. Graphically, this corresponds to the parabola intersecting the x-axis at two distinct points. The roots can be found using the quadratic formula, and they will be two different real numbers.

Case 2: $\Delta = 0$ (Zero Discriminant)

When the discriminant is zero, i.e., $b^2 - 4ac = 0$, the quadratic equation has one real root (a repeated root). This means there is exactly one real number that satisfies the equation, and it appears twice as a solution. Graphically, this corresponds to the parabola touching the x-axis at exactly one point (the vertex of the parabola lies on the x-axis). The root can be found using the quadratic formula, but since the square root part is zero, there is only one solution.

Case 3: $\Delta < 0$ (Negative Discriminant)

When the discriminant is negative, i.e., $b^2 - 4ac < 0$, the quadratic equation has two complex roots. This means there are no real number solutions; instead, the solutions involve complex numbers, which include an imaginary part (the imaginary unit $i$, where $i^2 = -1$). Graphically, this corresponds to the parabola not intersecting the x-axis at any point. The complex roots can be found using the quadratic formula, and they will be a pair of complex conjugates.

In summary, the discriminant is a powerful tool that provides valuable information about the roots of a quadratic equation without needing to solve the equation itself. By simply calculating $b^2 - 4ac$, we can determine whether the equation has two distinct real roots, one real root, or two complex roots.

Applying the Discriminant to $2x^2 + 5x - 8$

Now, let's apply our understanding of the discriminant to the specific polynomial $2x^2 + 5x - 8$. Our goal is to find the discriminant of this quadratic equation and interpret its value to understand the nature of its roots. To begin, we need to identify the coefficients $a$, $b$, and $c$ from the equation, which is in the standard form $ax^2 + bx + c = 0$.

In the equation $2x^2 + 5x - 8$, we have:

• $a = 2$ (the coefficient of the $x^2$ term)
• $b = 5$ (the coefficient of the $x$ term)
• $c = -8$ (the constant term)

Now that we have identified the coefficients, we can use the discriminant formula, which is:

$\Delta = b^2 - 4ac$

Substitute the values of $a$, $b$, and $c$ into the formula:

$\Delta = (5)^2 - 4(2)(-8)$

Now, we perform the calculations:

$\Delta = 25 - 4(-16)$

$\Delta = 25 + 64$

$\Delta = 89$

So, the discriminant of the polynomial $2x^2 + 5x - 8$ is 89. Now, let's interpret this value.

Determining the Discriminant of $2x^2+5x-8$

In this section, we will explicitly determine the discriminant of the quadratic polynomial $2x^2 + 5x - 8$. As we've discussed, the discriminant is a critical value that provides insight into the nature of the roots of a quadratic equation. We've already set the stage by identifying the coefficients and the discriminant formula. Now, we will walk through the calculation step-by-step to arrive at the discriminant's value.

We have the quadratic equation $2x^2 + 5x - 8$. From this equation, we identify the coefficients as follows:

• $a = 2$
• $b = 5$
• $c = -8$

Next, we use the discriminant formula:

$\Delta = b^2 - 4ac$

Now, we substitute the values of $a$, $b$, and $c$ into the formula:

$\Delta = (5)^2 - 4(2)(-8)$

This is where we carefully perform the arithmetic operations. First, we square $b$ (which is 5):

$\Delta = 25 - 4(2)(-8)$

Next, we multiply the numbers within the second term:

$\Delta = 25 - 4(-16)$

Now, we multiply -4 by -16:

$\Delta = 25 + 64$

Finally, we add 25 and 64:

$\Delta = 89$

Therefore, the discriminant of the quadratic polynomial $2x^2 + 5x - 8$ is 89. This calculation confirms the numerical value we obtained earlier, and now we are ready to interpret what this value means for the roots of the equation.

Interpreting the Discriminant of $2x^2 + 5x - 8$

Having calculated the discriminant of the polynomial $2x^2 + 5x - 8$, which is 89, we now turn to interpreting what this value tells us about the nature of the roots of the equation. The discriminant, $\Delta = 89$, is a positive number. As we learned earlier, the sign of the discriminant is crucial in determining whether the quadratic equation has two distinct real roots, one real root (a repeated root), or two complex roots.

Since $\Delta = 89 > 0$, we can conclude that the quadratic equation $2x^2 + 5x - 8 = 0$ has two distinct real roots. This means there are two different real numbers that satisfy the equation. Graphically, this implies that the parabola representing the quadratic equation will intersect the x-axis at two distinct points. These points correspond to the two real roots of the equation.

To find the exact values of the roots, we would use the quadratic formula:

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

In our case, $a = 2$, $b = 5$, and $c = -8$. Substituting these values into the quadratic formula, we get:

$x = \frac{-5 \pm \sqrt{89}}{4}$

This gives us two distinct real roots:

$x_1 = \frac{-5 + \sqrt{89}}{4}$ and $x_2 = \frac{-5 - \sqrt{89}}{4}$

These are the two points where the parabola intersects the x-axis. The discriminant has allowed us to predict the existence of these two real roots without having to go through the entire process of solving the equation. This demonstrates the power and utility of the discriminant as a tool for analyzing quadratic equations.

Conclusion

In this comprehensive guide, we've explored the concept of the discriminant in the context of quadratic polynomials. We've defined the discriminant, learned its formula ($\Delta = b^2 - 4ac$), and, most importantly, understood how to interpret its value. The discriminant is a fundamental tool for analyzing quadratic equations, providing valuable information about the nature of their roots without the need to solve the equations explicitly.

We've seen that:

• If $\Delta > 0$, the quadratic equation has two distinct real roots.
• If $\Delta = 0$, the quadratic equation has one real root (a repeated root).
• If $\Delta < 0$, the quadratic equation has two complex roots.

We applied this knowledge to the specific polynomial $2x^2 + 5x - 8$. By identifying the coefficients $a$, $b$, and $c$, we calculated the discriminant to be 89. Since 89 is a positive number, we concluded that the equation $2x^2 + 5x - 8 = 0$ has two distinct real roots. This means there are two different real numbers that satisfy the equation, and the parabola representing the equation intersects the x-axis at two points.

Understanding the discriminant is essential for anyone studying algebra and beyond. It allows for a quick and efficient analysis of quadratic equations, saving time and providing valuable insights into the nature of their solutions. The discriminant is a powerful tool in the mathematician's toolkit, enabling a deeper understanding of quadratic equations and their applications in various fields.