Given That One Of The Roots Of X 2 + hx10=0 Is 2 . (a) Find The Value Of Second Root. (b) Find The Value
Understanding quadratic equations is fundamental in algebra, and they frequently appear in various mathematical and real-world contexts. In this article, we will analyze the quadratic equation \(X^2 + hX + 10 = 0\), given that one of its roots is 2. We will explore how to find the second root and determine the specific value of the parameter \(h\) involved in the equation. This comprehensive guide aims to clarify these concepts with step-by-step solutions, explanations, and relevant formulas to enhance your understanding.
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Introduction to Quadratic Equations
A quadratic equation is a second-degree polynomial equation of the form:
\[
ax^2 + bx + c = 0
\]
where:
- \(a \neq 0\),
- \(b\) and \(c\) are constants.
The solutions, or roots, of the quadratic are the values of \(x\) that satisfy the equation. These roots can be real or complex, and their nature is determined by the discriminant:
\[
D = b^2 - 4ac
\]
- If \(D > 0\), there are two distinct real roots.
- If \(D = 0\), there is one real root (a repeated root).
- If \(D < 0\), the roots are complex conjugates.
Quadratic equations can be solved using various methods, including factoring, completing the square, and applying the quadratic formula:
\[
x = \frac{-b \pm \sqrt{D}}{2a}
\]
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Given Data and Problem Statement
The quadratic equation under consideration is:
\[
X^2 + hX + 10 = 0
\]
It is given that one of its roots is 2. The goal is to:
- (a) Find the second root.
- (b) Find the value of \(h\).
This problem involves understanding the relationships between roots and coefficients of quadratic equations, particularly Vieta's formulas.
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Understanding Vieta's Formulas
Vieta's formulas relate the roots of a quadratic equation to its coefficients:
- Sum of roots:
\[
\alpha + \beta = -\frac{b}{a}
\]
- Product of roots:
\[
\alpha \beta = \frac{c}{a}
\]
In our case:
\[
a = 1, \quad b = h, \quad c = 10
\]
Thus:
\[
\alpha + \beta = -h
\]
\[
\alpha \beta = 10
\]
Given that one root \(\alpha = 2\), we can leverage these relationships to find the second root \(\beta\) and the value of \(h\).
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Part (a): Find the Second Root
Since one root \(\alpha = 2\), and using the product of roots:
\[
\alpha \beta = 10
\]
Substitute \(\alpha = 2\):
\[
2 \times \beta = 10
\]
Solve for \(\beta\):
\[
\beta = \frac{10}{2} = 5
\]
Therefore, the second root is \(\boxed{5}\).
---
Part (b): Find the Value of \(h\)
Recall from Vieta's formulas:
\[
\alpha + \beta = -h
\]
Substitute the known roots:
\[
2 + 5 = -h
\]
\[
7 = -h
\]
Solve for \(h\):
\[
h = -7
\]
Thus, the value of \(h\) is \(\boxed{-7}\).
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Summary of Results
| Parameter | Value |
|------------|---------|
| Second root \(\beta\) | 5 |
| Coefficient \(h\) | -7 |
The quadratic equation with these parameters is:
\[
X^2 - 7X + 10 = 0
\]
which indeed has roots 2 and 5, satisfying the initial condition.
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Verification of Solutions
To ensure our solutions are correct, let's verify the roots by substituting them back into the quadratic equation:
\[
X^2 - 7X + 10 = 0
\]
- For \(X = 2\):
\[
(2)^2 - 7 \times 2 + 10 = 4 - 14 + 10 = 0
\]
- For \(X = 5\):
\[
(5)^2 - 7 \times 5 + 10 = 25 - 35 + 10 = 0
\]
Both roots satisfy the equation, confirming the correctness of our calculations.
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Additional Insights and Applications
Understanding how to find roots and coefficients in quadratic equations has numerous applications:
- Physics: Calculating projectile trajectories involves quadratic equations.
- Engineering: Structural analysis often requires solving quadratic equations for stress and strain.
- Economics: Profit maximization models sometimes involve quadratic functions.
Moreover, mastering Vieta's formulas provides an efficient way to analyze quadratics without fully solving them, especially when roots are partially known.
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Conclusion
In this detailed analysis, we've established a method to determine the second root and the parameter \(h\) for the quadratic equation \(X^2 + hX + 10 = 0\), given that one root is 2. By leveraging Vieta's formulas, we deduced:
- The second root is 5.
- The value of \(h\) is -7.
These results not only solve the specific problem but also reinforce fundamental algebraic principles applicable across various mathematical and scientific disciplines.
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Further Practice Problems
To solidify your understanding, consider practicing these problems:
- If one root of \(x^2 + 4x + k = 0\) is \(-3\), find the other root and the value of \(k\).
- For the quadratic equation \(2x^2 + h x + 8 = 0\), given that one root is 4, determine the other root and the value of \(h\).
- Given the quadratic \(x^2 + 6x + 9 = 0\), verify its roots and discuss their nature (real or complex).
Engaging with such problems enhances problem-solving skills and deepens understanding of quadratic equations.
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In summary, understanding the relationship between roots and coefficients using Vieta's formulas enables efficient solving of quadratic equations, especially when some roots are known. The methodology demonstrated here can be applied broadly to similar algebraic problems, reinforcing the importance of foundational algebra in various fields.
