Quadratic Equation Whose Roots Are 3 And 4

The quadratic equation whose rootsare 3 and 4 is a fundamental concept in algebra, illustrating the direct relationship between the solutions of an equation and its coefficients. Understanding how to construct such an equation from given roots is a crucial skill for students and professionals alike. This article will guide you through the process step-by-step, explain the underlying mathematics, and address common questions.

Introduction

A quadratic equation is a second-degree polynomial equation, typically written in the form ( ax^2 + bx + c = 0 ), where ( a ), ( b ), and ( c ) are constants, and ( a \neq 0 ). The solutions to this equation are called its roots. For instance, consider a quadratic equation where the roots are given as 3 and 4. Knowing these roots allows us to reconstruct the original equation. This article will demonstrate precisely how to achieve this, providing a clear, methodical approach that leverages the sum and product of the roots. Mastering this technique is essential for solving more complex problems involving quadratic functions and their graphs.

Steps to Form the Quadratic Equation

To construct a quadratic equation with specific roots, follow these systematic steps:

1. Identify the Roots: Clearly state the given roots. For this example, the roots are 3 and 4.
2. Calculate the Sum of Roots: Add the two roots together. Here, ( 3 + 4 = 7 ).
3. Calculate the Product of Roots: Multiply the two roots together. Here, ( 3 \times 4 = 12 ).
4. Form the Equation: Using the standard form ( x^2 - (\text{Sum of Roots})x + (\text{Product of Roots}) = 0 ), substitute the values obtained. This gives ( x^2 - 7x + 12 = 0 ).

Scientific Explanation

The derivation of the quadratic equation from its roots relies on the fundamental properties of polynomials. If ( r_1 ) and ( r_2 ) are the roots of a quadratic equation ( ax^2 + bx + c = 0 ), then the equation can be expressed as ( a(x - r_1)(x - r_2) = 0 ). Expanding this product:

( a(x - r_1)(x - r_2) = a[ x^2 - (r_1 + r_2)x + (r_1 \times r_2) ] = ax^2 - a(r_1 + r_2)x + a(r_1 \times r_2) )

For simplicity and convention, we often set ( a = 1 ), resulting in the monic quadratic equation:

( x^2 - (r_1 + r_2)x + (r_1 \times r_2) = 0 )

This formula directly links the sum and product of the roots to the coefficients of the equation. In our case, ( r_1 = 3 ), ( r_2 = 4 ), so ( r_1 + r_2 = 7 ) and ( r_1 \times r_2 = 12 ), leading to ( x^2 - 7x + 12 = 0 ).

Verification

To confirm that the equation ( x^2 - 7x + 12 = 0 ) indeed has roots 3 and 4, we can factor it:

( x^2 - 7x + 12 = (x - 3)(x - 4) = 0 )

Setting each factor to zero gives ( x - 3 = 0 ) or ( x - 4 = 0 ), confirming the roots are 3 and 4. Alternatively, using the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) with ( a=1, b=-7, c=12 ):

( x = \frac{7 \pm \sqrt{49 - 48}}{2} = \frac{7 \pm 1}{2} ), yielding ( x = 4 ) or ( x = 3 ).

Frequently Asked Questions (FAQ)

• Q: Can the quadratic equation have complex roots? Yes, if the discriminant (( b^2 - 4ac )) is negative, the roots will be complex conjugates. However, for real roots like 3 and 4, the discriminant is positive.
• Q: What if the roots are repeated? If the roots are identical (e.g., both 3), the equation becomes ( (x - 3)^2 = 0 ) or ( x^2 - 6x + 9 = 0 ).
• Q: Can I use this method for higher-degree polynomials? This specific method applies directly to quadratic equations (degree 2). For cubic or higher-degree polynomials, different techniques are required.
• Q: Why do we set ( a = 1 )? Setting ( a = 1 ) gives a monic quadratic equation, which is often the simplest and most common form used. The method works regardless of ( a )'s value, but the sum and product formulas adjust accordingly.
• Q: How is this useful in real life? Understanding roots helps in modeling real-world scenarios like projectile motion (where roots represent launch and landing points), economics (break-even analysis), and engineering (stability analysis).

Conclusion

Constructing a quadratic equation from its given roots is a straightforward process grounded in the relationship between the roots and the coefficients. By identifying the roots, calculating their sum and product, and applying the standard form ( x^2 - (\text{Sum})x + (\text{Product}) = 0 ), you can efficiently derive the equation. This foundational skill not only reinforces algebraic manipulation but also provides a powerful tool for analyzing quadratic functions and their applications across various disciplines. Mastery of this concept paves the way for tackling more advanced mathematical challenges.

Mastering the technique of deriving quadraticequations from given roots is not merely an algebraic exercise; it is a fundamental skill that bridges abstract mathematics with tangible real-world applications. The process, rooted in the intrinsic relationship between roots and coefficients, provides a powerful method for modeling phenomena ranging from projectile motion to financial break-even points. By systematically applying