Solve Quadratic Equations: Discriminant Method

When tackling quadratic equations, a powerful tool at our disposal is the discriminant. This mathematical gem allows us to peek into the nature of the solutions without actually solving for them. For the equation $x^2 + 15x + 13 = 0$, understanding the discriminant is key to unlocking the secrets of its roots. Let's dive deep into how we can compute this discriminant and, from there, unveil the number and types of solutions this specific equation holds. The process is straightforward, and by the end, you'll be able to approach similar problems with confidence. We'll break down each step, ensuring clarity and a solid grasp of the underlying principles. The quadratic formula, a cornerstone of algebra, is intimately linked with the discriminant, and exploring their relationship will only deepen your understanding.

Understanding the Discriminant

The discriminant is a vital part of the quadratic formula, which is used to solve equations of the form $ax^2 + bx + c = 0$. The discriminant is represented by the Greek letter delta ($\Delta$) and is calculated as $\Delta = b^2 - 4ac$. This single value, derived from the coefficients of the quadratic equation, tells us a tremendous amount about the solutions we can expect. It's like a predictor, giving us a preview of what lies ahead. The beauty of the discriminant lies in its simplicity and the wealth of information it provides. Whether the solutions are real and distinct, real and equal, or complex conjugates, the discriminant reveals all. This is particularly useful when we only need to know the nature of the solutions, saving us the time and effort of a full computation. We will focus on how this formula is applied to our specific problem $x^2 + 15x + 13 = 0$. Here, $a = 1$, $b = 15$, and $c = 13$. Plugging these values into the discriminant formula will be our first major step. The value of $a$, $b$, and $c$ are crucial here and directly impact the outcome of the discriminant calculation. It's important to correctly identify these coefficients from the given quadratic equation to avoid any errors in the subsequent steps. The formula itself is elegant in its construction, using a simple subtraction of a product from a squared term, yet its implications are profound in the realm of quadratic analysis.

Calculating the Discriminant for $x^2+15 x+13=0$

Now, let's get down to business and calculate the discriminant for our equation, $x^2 + 15x + 13 = 0$. As identified earlier, the coefficients are $a=1$, $b=15$, and $c=13$. Substituting these values into the discriminant formula, $\Delta = b^2 - 4ac$, we get: $\Delta = (15)^2 - 4(1)(13)$. First, we square $b$: $15^2 = 225$. Then, we calculate the product of $4ac$: $4 \times 1 \times 13 = 52$. Finally, we subtract the second result from the first: $\Delta = 225 - 52$. Performing this subtraction, we find that $\Delta = 173$. This computed value, 173, is the discriminant for the given quadratic equation. It's a positive number, and this characteristic is the first clue to the nature of our solutions. The calculation was precise, and the result is definitive. Each component of the formula played its part, and the final number, 173, stands as a testament to the power of this algebraic tool. The straightforward substitution and arithmetic operations ensure that this step is accessible to anyone familiar with basic algebra. Remember, the signs of the coefficients are important; a negative coefficient would alter the calculation of $4ac$. In this case, all coefficients are positive, simplifying the arithmetic. The magnitude of the discriminant, 173, is also significant, indicating a substantial difference between the squared term and the product term.

Determining the Number and Type of Solutions

With the discriminant calculated as $\Delta = 173$, we can now determine the number and type of solutions for $x^2 + 15x + 13 = 0$. The value of the discriminant dictates these properties:

• If $\Delta > 0$: The equation has two distinct real solutions. This means there are two different numbers that satisfy the equation.
• If $\Delta = 0$: The equation has exactly one real solution (or two equal real solutions). This occurs when the quadratic is a perfect square trinomial.
• If $\Delta < 0$: The equation has two complex conjugate solutions. These solutions involve the imaginary unit 'i'.

In our case, $\Delta = 173$. Since 173 is greater than 0, we fall into the first category. Therefore, the equation $x^2 + 15x + 13 = 0$ has two distinct real solutions. These solutions will be irrational because 173 is not a perfect square, but they are definitely real numbers. This classification is a direct consequence of the positive value of the discriminant. The fact that the discriminant is positive implies that the parabola represented by the quadratic equation intersects the x-axis at two distinct points. These intersection points correspond to the real roots of the equation. The type of real solution (rational or irrational) depends further on whether the discriminant is a perfect square. Since 173 is not a perfect square (e.g., $13^2 = 169$ and $14^2 = 196$), the solutions will be irrational. However, the primary classification remains that they are two distinct real numbers. This understanding allows us to predict the outcome without performing the full quadratic formula calculation, which would yield the exact values of these solutions. The discriminant acts as a powerful filter, giving us immediate insight into the nature of the roots. The positive value is the key indicator here, and its magnitude further informs us about the separation between these two real roots.

Exploring the Solutions Further (Optional)

While the discriminant has already told us that we have two distinct real solutions, we can optionally use the quadratic formula to find their exact values. The quadratic formula is $x = \frac{-b \pm \sqrt{\Delta}}{2a}$. Plugging in our values $a=1$, $b=15$, and $\Delta=173$, we get: $x = \frac{-15 \pm \sqrt{173}}{2(1)}$. This simplifies to $x = \frac{-15 \pm \sqrt{173}}{2}$. So, the two distinct real solutions are $x_1 = \frac{-15 + \sqrt{173}}{2}$ and $x_2 = \frac{-15 - \sqrt{173}}{2}$. As predicted by the discriminant not being a perfect square, these are indeed irrational numbers. This step confirms the information provided by the discriminant and provides the concrete values of the roots. It's a good practice to perform this step if the exact solutions are required, or to verify the nature of the roots determined by the discriminant. The presence of $\sqrt{173}$ indicates their irrationality. The $\pm$ sign signifies the two distinct solutions. The denominator of $2a$ ensures we are scaling correctly according to the quadratic formula's derivation. This optional step reinforces the power and utility of the discriminant by showing how it aligns perfectly with the results obtained from the full solution method. It's a valuable check and provides a complete picture of the equation's roots.

Conclusion

In summary, the discriminant is an indispensable tool for analyzing quadratic equations. For the equation $x^2 + 15x + 13 = 0$, we successfully computed the discriminant to be $\Delta = 173$. Because this value is positive ($\Delta > 0$), we definitively determined that the equation possesses two distinct real solutions. This method not only provides quick insights into the nature of the roots but also saves considerable time when the actual values of the roots are not immediately necessary. Mastering the concept of the discriminant enhances your problem-solving capabilities in algebra significantly. It's a fundamental concept that underpins much of our understanding of quadratic functions and their graphical representations. Remember, the sign of the discriminant is the key; positive means two real and distinct, zero means one real (or two equal), and negative means two complex. This principle holds true for all quadratic equations.

For further exploration into quadratic equations and their properties, you can visit Khan Academy or Wolfram MathWorld.