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Chapter 11: Problem 41

Write a quadratic equation having the given numbers as solutions. $$ -\sqrt{3}, \sqrt{3} $$

Short Answer

Expert verified
The quadratic equation is \( x^2 - 3 = 0 \).

Step by step solution

01

- Understand the Given Solutions

The given solutions of the quadratic equation are \(-\sqrt{3}\) and \(\sqrt{3}\).
02

- Recall the Form of a Quadratic Equation

A quadratic equation with solutions \(p\) and \(q\) can be written in the form: \((x - p)(x - q) = 0\).
03

- Substitute the Given Solutions

Substitute the given solutions into the quadratic form. So the equation will be: \( (x - (-\sqrt{3}))(x - (\sqrt{3})) = 0 \).
04

- Simplify the Equation

Simplify the expression: \( (x + \sqrt{3})(x - \sqrt{3}) = 0 \).
05

- Expand the Terms

Expand the product using the difference of squares formula \((a+b)(a-b) = a^2 - b^2\): \( x^2 - (\sqrt{3})^2 = 0 \).
06

- Write Final Equation

Since \( (\sqrt{3})^2 = 3 \), the final quadratic equation becomes: \ x^2 - 3 = 0 \.

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Key Concepts

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

Solutions of Quadratic Equations
A quadratic equation can have either real or complex solutions. In this exercise, we are given the solutions as \(- \sqrt{3}\) and \(\sqrt{3}\). A quadratic equation is generally written as \(ax^2 + bx + c = 0\). When given the solutions directly, you can use the fact that they are the roots to write the equation in factored form: \( (x - p)(x - q) = 0 \), where \( p \) and \( q \) are the solutions. \In this case, substituting the solutions \(-\sqrt{3}\) and \(\sqrt{3}\), we get: \( (x + \sqrt{3})(x - \sqrt{3}) = 0 \). This equation showcases the key concept of generating a quadratic equation given its solutions. \
Difference of Squares Formula
The difference of squares formula is one of the most efficient tools to simplify quadratic expressions. It states that \( (a + b)(a - b) = a^2 - b^2 \). In this problem, we apply this formula to simplify \( (x + \sqrt{3})(x - \sqrt{3}) \). By substituting directly, we see: \( x^2 - (\sqrt{3})^2 \). \This technique is especially useful when dealing with quadratic equations that involve terms with squares. The difference of squares formula helps convert a factored form to a standard quadratic expression quickly and accurately.
Expanding Quadratic Expressions
Expanding quadratic expressions often involves multiplication of binomials. Using the difference of squares formula, we expand \( (x + \sqrt{3})(x - \sqrt{3}) \) to \( x^2 - (\sqrt{3})^2 \). \In this exercise, expanding the simplified equation \( x^2 - 3 = 0 \) shows how you move from a factored form to a quadratic expression. When you fully understand how to expand, you gain the ability to transform solutions into their corresponding quadratic equations, aiding in various algebraic problems. The expanding process is essential to reach the final equation, where we find that \( (\sqrt{3})^2 \) equals 3, thus giving us the simplified form \ x^2 - 3 = 0 \.

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