Problem 83 - A Graphical Approach to Precalculus with Limits

Question

For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverses $$2-\sqrt{5}, 2+\sqrt{5}$$

Step-by-Step Solution

Answer

The values are \(a = 1\), \(b = -4\), and \(c = -1\).

Step 1: Understand the Concept of Roots

For a quadratic equation of the form \( ax^2 + bx + c = 0 \), if it has roots \( p \) and \( q \), it can be expressed as \((x - p)(x - q) = 0\). The given numbers are \(2 - \sqrt{5}\) and \(2 + \sqrt{5}\), which imply these are the roots of the equation.

Step 2: Formulate the Quadratic Equation

Rewrite the equation based on its roots: \((x - (2 - \sqrt{5}))(x - (2 + \sqrt{5})) = 0\). This step involves expressing the equation in the factorized form indicating these are the roots.

Step 3: Expand the Factors

Expand the product \((x - (2 - \sqrt{5}))(x - (2 + \sqrt{5}))\) using the formula for the difference of squares, since \((a - b)(a + b) = a^2 - b^2\). Here, \(a = x - 2\) and \(b = \sqrt{5}\).

Step 4: Simplify the Equation

Calculate this further as:\[(x - (2 - \sqrt{5}))(x - (2 + \sqrt{5})) = (x - 2)^2 - (\sqrt{5})^2\]Simplifying each part gives:\[(x - 2)^2 = x^2 - 4x + 4\]and\[ (\sqrt{5})^2 = 5\].So, substituting back:\[ x^2 - 4x + 4 - 5 = x^2 - 4x - 1 \].

Step 5: Identify the Coefficients

Now, compare \(x^2 - 4x - 1 = 0\) with the general quadratic equation form \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = -4\), and \(c = -1\).

Key Concepts

Zero-Product Property

The Zero-Product Property is a powerful tool in algebra, especially when dealing with quadratic equations. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. This can be symbolically written as:

• If \( ab = 0 \), then \( a = 0 \) or \( b = 0 \).

This property helps in solving factorized equations. When a quadratic equation is presented in its factorized form, like \((x - p)(x - q) = 0\), you can directly state its solutions, or roots, as \( x = p \) and \( x = q \). This is the basic principle behind solving quadratics by factoring.Understanding this property is essential for unraveling more complex quadratic problems and helps in re-expressing quadratic equations in a meaningful way.

Roots of a Quadratic Equation

The roots of a quadratic equation are, simply put, the solutions for which the equation equals zero. When dealing with a quadratic equation like \( ax^2 + bx + c = 0 \), the roots are the values of \( x \) that satisfy this equation. There are several ways to determine these roots:

• Factorization: Express the equation in the form \((x - p)(x - q) = 0\) using the known roots \( p \) and \( q \).
• Quadratic formula: This use the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to calculate the roots.

In our exercise, we directly used the concept of roots from the given values, \(2 - \sqrt{5}\) and \(2 + \sqrt{5}\), indicating that these are the values of \( x \) which make the quadratic equation zero. By recognizing the roots, we can reconstruct the entire equation.

Difference of Squares

The difference of squares is a special algebraic identity that is extremely useful when expanding expressions. It is given by the formula: \((a - b)(a + b) = a^2 - b^2\). This identity allows us to easily expand products of two conjugates.

• For example, in our exercise: \((x - (2 - \sqrt{5}))(x - (2 + \sqrt{5}))\) was transformed using \( a = x - 2 \) and \( b = \sqrt{5} \).
• This resulted in \((x - 2)^2 - (\sqrt{5})^2\), which simplifies to \(x^2 - 4x - 1\).

This simplification makes use of the identity to make complex multiplications straightforward and is crucial when forming or solving quadratic equations through factoring.

Coefficient Identification

When you have the expanded form of a quadratic equation, identifying its coefficients is key. In general, a quadratic equation is written as \( ax^2 + bx + c = 0 \), and each letter \( a \), \( b \), and \( c \) corresponds to the coefficients.

• In the problem at hand, after the simplification, the equation becomes \( x^2 - 4x - 1 = 0 \).
• Here, \( a = 1 \) (as there's no coefficient other than 1 in front of \( x^2 \)), \( b = -4 \), and \( c = -1 \).

Identifying these coefficients correctly is crucial for applying further operations or verifying if the quadratic equation matches specific criteria or roots. Through coefficient identification, you can confirm or contest the solutions to the problem at hand.