Algebraic Equations And Roots Finding Values And Analyzing Relationships

In the realm of mathematics, algebraic equations and the relationships between their roots form the bedrock of countless problem-solving techniques. This article delves into two intriguing problems, dissecting their solutions and illuminating the underlying mathematical principles. We'll explore how to simplify radical expressions, manipulate equations, and apply fundamental algebraic concepts to arrive at accurate solutions. This journey will not only enhance your problem-solving skills but also deepen your understanding of the elegant interplay between algebra and equation analysis.

1. Simplifying Radical Expressions and Finding the Value of $\left(a + \frac{1}{a}\right)$

In this section, we tackle the problem of simplifying a radical expression and then finding the value of a specific expression involving the simplified form. Let's break down the problem step by step:

Understanding the Challenge

The core challenge lies in the initial radical expression: $a = \sqrt{7 + 4\sqrt{3}}$. The presence of a nested radical makes it difficult to directly compute expressions involving a. Therefore, our primary goal is to simplify this radical expression into a more manageable form. Once we've simplified a, we can easily calculate $\frac{1}{a}$ and, subsequently, the value of $a + \frac{1}{a}$.

The Simplification Process

The key to simplifying nested radicals of the form $\sqrt{x + y\sqrt{z}}$ often involves attempting to express the term inside the square root as a perfect square. In our case, we aim to rewrite $7 + 4\sqrt{3}$ in the form $(m + n\sqrt{3})^2$, where m and n are integers. Expanding $(m + n\sqrt{3})^2$, we get:

$(m + n\sqrt{3})^2 = m^2 + 2mn\sqrt{3} + 3n^2$

Now, we need to find integers m and n such that:

• $m^2 + 3n^2 = 7$
• $2mn = 4$

From the second equation, we get $mn = 2$. Considering integer values, possible pairs for (m, n) are (1, 2) and (2, 1). Let's test these pairs in the first equation:

• If (m, n) = (1, 2), then $1^2 + 3(2^2) = 1 + 12 = 13$, which doesn't equal 7.
• If (m, n) = (2, 1), then $2^2 + 3(1^2) = 4 + 3 = 7$, which satisfies the first equation.

Thus, we have found that m = 2 and n = 1. Therefore, we can rewrite the expression as:

$7 + 4\sqrt{3} = (2 + \sqrt{3})^2$

Taking the square root of both sides, we get:

$a = \sqrt{7 + 4\sqrt{3}} = \sqrt{(2 + \sqrt{3})^2} = 2 + \sqrt{3}$

Finding $\frac{1}{a}$

Now that we have simplified a, we can find its reciprocal:

$\frac{1}{a} = \frac{1}{2 + \sqrt{3}}$

To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator, which is $2 - \sqrt{3}$:

$\frac{1}{a} = \frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{2^2 - (\sqrt{3})^2} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}$

Calculating $a + \frac{1}{a}$

Finally, we can compute the value of $a + \frac{1}{a}$:

$a + \frac{1}{a} = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 2 + \sqrt{3} + 2 - \sqrt{3} = 4$

Therefore, the value of $a + \frac{1}{a}$ is 4. The correct answer is (b).

Key Takeaways

• Simplifying nested radicals often involves expressing the radicand as a perfect square.
• Rationalizing the denominator is crucial when dealing with fractions containing radicals.
• Careful algebraic manipulation is essential for arriving at the correct solution.

2. Analyzing Quadratic Equations and Equal Roots

This section focuses on a quadratic equation and the condition for its roots to be equal. We'll explore how the discriminant of a quadratic equation plays a vital role in determining the nature of its roots.

The Quadratic Equation and the Discriminant

We are given the quadratic equation:

$\left(a^2 + b^2\right)x^2 - 2(ac + bd)x + \left(c^2 + d^2\right) = 0$

For a quadratic equation of the form $Ax^2 + Bx + C = 0$, the discriminant, denoted by Δ (Delta), is given by:

$Δ = B^2 - 4AC$

The discriminant provides valuable information about the nature of the roots:

• If Δ > 0, the equation has two distinct real roots.
• If Δ = 0, the equation has two equal real roots (a repeated root).
• If Δ < 0, the equation has two complex roots.

In our case, we are told that the roots are equal. This means that the discriminant of the given quadratic equation must be equal to zero.

Applying the Discriminant Condition

Let's identify the coefficients A, B, and C in our equation:

• A = $a^2 + b^2$
• B = -2(ac + bd)
• C = $c^2 + d^2$

Now, we can calculate the discriminant:

$Δ = B^2 - 4AC = [-2(ac + bd)]^2 - 4(a^2 + b^2)(c^2 + d^2)$

Since the roots are equal, we set Δ = 0:

$0 = 4(ac + bd)^2 - 4(a^2 + b^2)(c^2 + d^2)$

Dividing both sides by 4, we get:

$0 = (ac + bd)^2 - (a^2 + b^2)(c^2 + d^2)$

Expanding both sides, we have:

$0 = a^2c^2 + 2acbd + b^2d^2 - (a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2)$

Simplifying the equation, we get:

$0 = a^2c^2 + 2abcd + b^2d^2 - a^2c^2 - a^2d^2 - b^2c^2 - b^2d^2$

$0 = 2abcd - a^2d^2 - b^2c^2$

Rearranging the terms, we obtain:

$a^2d^2 - 2abcd + b^2c^2 = 0$

Recognizing a Perfect Square

The left-hand side of the equation is a perfect square trinomial:

$(ad - bc)^2 = 0$

Taking the square root of both sides, we get:

$ad - bc = 0$

Therefore, the condition for the roots to be equal is:

$ad = bc$

Which can be rewritten as:

$\frac{a}{b} = \frac{c}{d}$

Conclusion

The relationship between the coefficients when the roots are equal is that a/b = c/d. This result highlights the crucial role of the discriminant in determining the nature of quadratic equation roots and the connection between the coefficients when specific root conditions are met.

Key Takeaways

• The discriminant of a quadratic equation determines the nature of its roots.
• Equal roots imply a discriminant of zero.
• Algebraic manipulation and pattern recognition (like perfect square trinomials) are essential for solving such problems.

Mastering Algebraic Equations and Root Analysis a Recap

Through these two problems, we've explored the importance of simplifying radical expressions and analyzing the discriminant of quadratic equations. These are fundamental concepts in algebra that are crucial for solving a wide range of mathematical problems. Remember to practice regularly, pay attention to algebraic manipulations, and understand the underlying principles. By doing so, you'll build a strong foundation in mathematics and enhance your problem-solving abilities.

Final Thoughts

Mathematics is not just about memorizing formulas; it's about understanding the relationships between concepts and applying them creatively to solve problems. By dissecting problems, understanding the underlying principles, and practicing consistently, you can unlock the beauty and power of mathematics. Whether it's simplifying complex expressions or analyzing the nature of roots, the journey of mathematical exploration is a rewarding one.