Problem 65 If we assume that a quadratic eq... [FREE SOLUTION]

Chapter 11: Problem 65

If we assume that a quadratic equation has integers for coefficients, will the product of the solutions always be a real number? Why or why not?

Short Answer

Expert verified

Yes, because the product simplifies to a rational (real) number.

Step by step solution

Understand the General Form of a Quadratic Equation

A quadratic equation with integer coefficients can be represented as \[ ax^2 + bx + c = 0 \]. Here, \( a \), \( b \), and \( c \) are integers.

Use Vieta's Formulas

By Vieta's formulas, for the quadratic equation \( ax^2 + bx + c = 0 \), the product of the solutions \( x_1 \) and \( x_2 \) is given by \( x_1 x_2 = \frac{c}{a} \).

Assess the Nature of the Product

Since both \( c \) and \( a \) are integers, the fraction \( \frac{c}{a} \) is a rational number. All rational numbers are also real numbers.

Conclusion

Thus, the product of the solutions of a quadratic equation with integer coefficients will always be a real number, as it simplifies to a rational number.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integer Coefficients

When we talk about **integer coefficients** in a quadratic equation, we mean that the numbers in front of the variables are whole numbers (both positive and negative, including zero). Such an equation can be written as \[ ax^2 + bx + c = 0 \]. Here,

**a**, **b**, and **c** are integer values.
The variable **x** represents the unknowns in the equation.

Since integers are real numbers, coefficients that are integers ensure that the quadratic formula interacts with real, whole number values to produce our solutions. In quadratic equations,

**a** cannot be zero (because otherwise, it wouldn't be quadratic).
**b** and **c** can be any integer, including zero.

This framework guarantees that while working with these equations, the properties of integers such as being rational and real inherently hold.

Vieta's Formulas

Vieta's formulas are an essential tool when considering the roots (or solutions) of polynomial equations. For a quadratic equation \[ ax^2 + bx + c = 0 \], Vieta's formulas tell us two crucial pieces of information:

The sum of the roots: \( x_1 + x_2 = -\frac{b}{a} \).
The product of the roots: \( x_1 x_2 = \frac{c}{a} \).

These relationships stem from polynomial factorization and are particularly useful. For instance, to determine the product of the roots in our case,

The product formula \( x_1 x_2 = \frac{c}{a} \) simplifies our computation.
Knowing **a** and **c** are integers, we see that their ratio forms a rational number.

Vieta's formulas thus serve as a bridge, making it easier to relate the coefficients of the quadratic equation directly to its roots without solving the equation entirely every time.

Real Number Solutions

A key question is whether the solutions (roots) to quadratic equations with integer coefficients result in **real numbers**. When we look at real numbers, we are referring to any number on the number line: including both rational and irrational numbers. Here's how we verify:

For quadratic equations, the discriminant \( \Delta = b^2 - 4ac \) determines the nature of the roots.
If \( \Delta \geq 0 \) (non-negative), the roots are real numbers. This encompasses both rational and irrational numbers.

When working specifically with integer coefficients and using Vieta's formulas, we inherently ensure:

The roots will multiply to form a rational number (since it is \( \frac{c}{a} \)).
Ratios of integers (rational numbers) are always real.

Hence, quadratic equations with integer coefficients guarantee, at the very least, that the product of their solutions is real. Real solutions ensure the consistency of real-world applications and further mathematical computations.

Rational Numbers

In discussing products of solutions, **rational numbers** play a vital role. A rational number is any number that can be expressed as the fraction \( \frac{p}{q} \), where **p** and **q** are integers and **q** is not zero. Here's how they connect to our quadratic equation:

From Vieta鈥檚 product formula, \( x_1 x_2 = \frac{c}{a} \), both **c** and **a** are integers.
The division of integers results in a rational number.

Rational numbers are crucial because they bridge the gap between the abstract nature of a quadratic鈥檚 structure and practical computation. Additionally,

Rational numbers are a subset of real numbers.
They provide exact values that are easier to manage in both pure and applied contexts.

This ensures when discussing quadratic equations with integer coefficients, the simplicity and exactness needed for accurate mathematical and real-world problem-solving are maintained.

91影视

If we assume that a quadratic equation has integers for coefficients, will the product of the solutions always be a real number? Why or why not?

Short Answer

Step by step solution

Understand the General Form of a Quadratic Equation

Use Vieta's Formulas

Assess the Nature of the Product

Conclusion

Key Concepts

Integer Coefficients

Vieta's Formulas

Real Number Solutions

Rational Numbers

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