Problem 79 - 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 $$1+\sqrt{2}, 1-\sqrt{2}$$

Step-by-Step Solution

Answer

The coefficients are \( a = 1 \), \( b = -2 \), \( c = -1 \).

Step 1: Identify the format of the quadratic equation

For a quadratic equation with roots \( r_1 \) and \( r_2 \), we can express it in the form \( a(x - r_1)(x - r_2) = 0 \). Here, the roots are \( 1 + \sqrt{2} \) and \( 1 - \sqrt{2} \).

Step 2: Express as a factored equation

Using the roots, express the equation as \( a(x - (1 + \sqrt{2}))(x - (1 - \sqrt{2})) = 0 \).

Step 3: Expand the factored form

Apply the distributive property to expand the expression: \[(x - 1 - \sqrt{2})(x - 1 + \sqrt{2}) = (x - 1)^2 - (\sqrt{2})^2\].

Step 4: Simplify the expression

Simplify the expanded expression: \[(x - 1)^2 - 2 = x^2 - 2x + 1 - 2 = x^2 - 2x - 1 \].

Step 5: Assign coefficients

In the simplified expression \( x^2 - 2x - 1 \), identify the coefficients: \( a = 1 \), \( b = -2 \), \( c = -1 \).

Key Concepts

Zero-Product Property

Understanding the Zero-Product Property is key when working with quadratic equations that are expressed in a factored form. This property states that if a product of numbers is zero, then at least one of the factors must be zero. In other words, if you have an equation like \[ a imes b = 0 \] e you can deduce that either \( a = 0 \) or \( b = 0 \), or possibly both.
This principle is especially useful when solving quadratic equations, because if a quadratic can be factorized into the form \[ a(x - r_1)(x - r_2) = 0 \] you can immediately state that \[ x - r_1 = 0 \] or \( x - r_2 = 0 \).
This gives you the roots \( x = r_1 \) and \( x = r_2 \).

• It makes finding solutions straightforward because the equation is already separated into simpler binomial expressions.
• When the quadratic is nicely factorable, it saves you the effort of using more complex methods like completing the square or the quadratic formula.

Instead, you leverage the zero-product property to quickly identify the values of \( x \) that satisfy the equation.

Factored Form

The act of expressing a quadratic equation in its factored form is an essential skill in algebra.
The factored form of a quadratic equation is \[ a(x - r_1)(x - r_2) = 0 \], where \( r_1 \) and \( r_2 \) are the roots of the quadratic equation.
To reach the factored form, you make use of known roots of the equation. For example, if you have roots \( 1 + \sqrt{2} \) and \( 1 - \sqrt{2} \), the factored form becomes \[ a(x - (1 + \sqrt{2}))(x - (1 - \sqrt{2})) = 0 \].
Converting a quadratic into factored form offers several advantages:

• It highlights the solutions directly, showing where the graph crosses the x-axis.
• It's an elegant solution approach, especially when the equation is easily factorable.
• Factored form often simplifies the process of graphing quadratics.

Achieving this form is easier when you start from known roots, as seen in this example, but can also be performed by factorizing complex quadratics through various algebraic methods.

Roots of Quadratic Equation

Roots of a quadratic equation refer to the values of \( x \) for which \[ ax^2 + bx + c = 0 \] holds true. These are essentially the x-values where the graph of the equation intersects the x-axis.
Finding the roots can be accomplished using different methods:

• Factoring, as described, involves expressing the equation in factored form.
• Completing the square converts the equation into a perfect square trinomial.
• Quadratic formula, which uses the formula \[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \] to find roots directly.

In this specific case, the equation's roots were given as \( 1 + \sqrt{2} \) and \( 1 - \sqrt{2} \). Using these, the factored version was constructed, and from there, checking the solutions becomes seamless. Understanding and finding roots is crucial because they provide key insights into the behavior of the quadratic function, such as its axis of symmetry and vertex, enriching your comprehension of the parabolic shape.