Solving Quadratic Equations A Step By Step Guide To Finding The Sum Of Roots

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In the realm of mathematics, quadratic equations hold a position of paramount importance. These equations, characterized by their second-degree polynomial form, arise in a myriad of applications, from modeling projectile motion to optimizing financial investments. Understanding the properties and solutions of quadratic equations is therefore crucial for any aspiring mathematician or scientist.

Delving into the Quadratic Equation

The quadratic equation we are presented with, $x^2 - 1x - 90 = 0$, is a classic example of this fundamental mathematical construct. Our goal is to decipher the solutions, often referred to as roots, of this equation. These roots, denoted as ${a, b}$, represent the values of $x$ that satisfy the equation, causing it to hold true.

Unlocking the Secrets of Roots: The crux of the problem lies in determining the sum of these roots, $a + b$. While we could embark on the traditional route of solving the quadratic equation to find the individual values of $a$ and $b$, there exists a more elegant and efficient approach that leverages the inherent properties of quadratic equations.

Harnessing Vieta's Formulas

Vieta's formulas, named after the renowned French mathematician François Viète, provide a powerful shortcut for determining the sum and product of the roots of a quadratic equation without explicitly solving for them. These formulas establish a direct relationship between the coefficients of the quadratic equation and its roots.

The Essence of Vieta's Formulas: For a general quadratic equation of the form $ax^2 + bx + c = 0$, Vieta's formulas state that:

  • The sum of the roots, $a + b$, is equal to $-b/a$.
  • The product of the roots, $a * b$, is equal to $c/a$.

Applying Vieta's Formulas to Our Equation: In our specific equation, $x^2 - 1x - 90 = 0$, we can identify the coefficients as follows:

  • a=1a = 1

  • b=−1b = -1

  • c=−90c = -90

Now, we can directly apply Vieta's formula for the sum of the roots:

a+b=−b/a=−(−1)/1=1a + b = -b/a = -(-1)/1 = 1

Therefore, the sum of the roots of the equation $x^2 - 1x - 90 = 0$ is 1. This corresponds to option C in the given choices.

Alternative Approach: Factoring the Quadratic

While Vieta's formulas provide a swift solution, it's worthwhile to explore an alternative method – factoring the quadratic equation. Factoring involves expressing the quadratic expression as a product of two linear expressions. This approach not only helps in finding the roots but also deepens our understanding of the equation's structure.

The Art of Factoring: To factor the quadratic $x^2 - 1x - 90$, we seek two numbers that multiply to $-90$ and add up to $-1$. After some contemplation, we can identify these numbers as $-10$ and $9$. Thus, we can rewrite the quadratic expression as:

x2−1x−90=(x−10)(x+9)x^2 - 1x - 90 = (x - 10)(x + 9)

Finding the Roots: Setting the factored expression equal to zero, we get:

(x−10)(x+9)=0(x - 10)(x + 9) = 0

This equation holds true if either $(x - 10) = 0$ or $(x + 9) = 0$. Solving these linear equations, we obtain the roots:

  • x=10x = 10

  • x=−9x = -9

Thus, the roots of the equation are ${10, -9}$. The sum of these roots is $10 + (-9) = 1$, which aligns perfectly with the result obtained using Vieta's formulas.

The Significance of the Discriminant

Before we conclude our exploration, let's briefly touch upon the concept of the discriminant. The discriminant, denoted as $\Delta$, is a crucial parameter that sheds light on the nature of the roots of a quadratic equation. It is defined as:

Δ=b2−4ac\Delta = b^2 - 4ac

Decoding the Discriminant: The value of the discriminant reveals whether the quadratic equation has:

  • Two distinct real roots (if $\Delta > 0$)
  • One real root (a repeated root) (if $\Delta = 0$)
  • No real roots (two complex roots) (if $\Delta < 0$)

Calculating the Discriminant for Our Equation: For our equation, $x^2 - 1x - 90 = 0$, the discriminant is:

Δ=(−1)2−4(1)(−90)=1+360=361\Delta = (-1)^2 - 4(1)(-90) = 1 + 360 = 361

Since $\Delta > 0$, we confirm that our equation has two distinct real roots, as we indeed found earlier.

Conclusion: A Symphony of Mathematical Concepts

In this exploration, we've successfully determined the sum of the roots of the quadratic equation $x^2 - 1x - 90 = 0$ using Vieta's formulas and the method of factoring. We've also touched upon the significance of the discriminant in understanding the nature of the roots. This journey underscores the interconnectedness of mathematical concepts and the power of employing diverse techniques to solve a single problem. Whether you're a student delving into the world of algebra or a seasoned mathematician seeking elegant solutions, the principles we've discussed here will undoubtedly prove invaluable in your mathematical endeavors. The answer, as we've definitively established, is 1. This exploration not only provides a solution but also illuminates the beauty and depth inherent in quadratic equations, inviting further exploration and discovery in the vast realm of mathematics.

Through this comprehensive analysis, we've not only answered the question but also delved into the underlying mathematical principles, enriching our understanding of quadratic equations and their solutions.