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Chapter 8: Problem 51

Solve. See Examples 1 through \(5 .\). $$ (x+3)\left(x^{2}-3 x+9\right)=0 $$

Short Answer

Expert verified
The solution is \(x = -3\). The quadratic has no real solutions.

Step by step solution

01

Identify the Zero Product Property

When an equation is set to zero, and it involves the product of factors, the Zero Product Property states that one or more of the factors must be zero for the product to be zero. Therefore, if \((x + 3)(x^2 - 3x + 9) = 0\), either \(x + 3 = 0\) or \(x^2 - 3x + 9 = 0\) or both must be true.
02

Solve the First Factor

Set the first factor \(x + 3 = 0\) and solve for \(x\). \[ x + 3 = 0 \]Subtract 3 from both sides to find:\[ x = -3 \]
03

Solve the Second Factor

Set the second factor \(x^2 - 3x + 9 = 0\) and solve for \(x\). Since it is a quadratic equation, we can use the quadratic formula where \(a = 1\), \(b = -3\), and \(c = 9\). The quadratic formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute the values:\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \times 1 \times 9}}{2 \times 1} \]\[ x = \frac{3 \pm \sqrt{9 - 36}}{2} \]\[ x = \frac{3 \pm \sqrt{-27}}{2} \]Since the discriminant is negative, \(-27\), this equation has no real solutions.

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Key Concepts

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

Zero Product Property
The Zero Product Property is a crucial concept in solving equations that involve the multiplication of factors. Essentially, it states that if you have a product of terms equal to zero, then at least one of the terms must be zero. For example, in the equation
  • \((x+3)(x^2-3x+9) = 0\),
either
  • \(x+3 = 0\)
  • or \(x^2-3x+9 = 0\)
  • or both need to be zero
Thus, by solving each factor separately, you can identify the potential values for \(x\) that satisfy the equation. This property allows complex problems to be broken into simpler, more manageable ones by considering each factor on its own.
Quadratic Formula
The quadratic formula provides a method to solve any quadratic equation. A quadratic equation is typically in the form
  • \(ax^2 + bx + c = 0\)
where \(a\), \(b\), and \(c\) are constants. The quadratic formula is:
  • \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula is incredibly useful as it applies universally to any quadratic equation, making it a fundamental tool in algebra. For example, in solving
  • \(x^2 - 3x + 9 = 0\),
we substitute \(a = 1\), \(b = -3\), and \(c = 9\) into the quadratic formula to find potential solutions.
Discriminant in Quadratic Equations
The discriminant is the part of the quadratic formula under the square root,
  • \(b^2 - 4ac\).
The value of the discriminant gives important information about the nature of the roots of a quadratic equation.
  • If the discriminant is positive, the equation has two distinct real roots.
  • If it is zero, there is exactly one real root, or a "double root".
  • If negative, as seen with \(x^2 - 3x + 9 = 0\), the equation has no real roots but instead two complex roots.
Here, the discriminant
  • \(-27\)
is negative, indicating no real solutions, which is a critical insight when using the quadratic formula for solving.

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Most popular questions from this chapter

A common equation used in business is a demand equation. It expresses the relationship between the unit price of some commodity and the quantity demanded. For Exercises III and 112 , p represents the unit price and x represents the quantity demanded in thousands. Acme, Inc., sells desk lamps and has found that the demand equation for a certain style of desk lamp is given by the equation \(p=-x^{2}+15 .\) Find the demand for the desk lamp if the price is \(\$ 7\) per lamp.

Solve. See Example 5 The Utah Ski Club sells calendars to raise money. The profit \(P,\) in cents, from selling \(x\) calendars is given by the equation \(P(x)=360 x-x^{2}\) A. Find how many calendars must be sold to maximize profit. B. Find the maximum profit.

Without calculating, tell whether each graph has a minimum value or a maximum value. See the Concept Check in the section. $$ f(x)=3-\frac{1}{2} x^{2} $$

Find two possible missing terms so that each is a perfect square trinomial. $$ z^{2}+\quad+\frac{25}{4} $$

Without solving, determine whether the solutions of each equation are real numbers or complex but not real numbers. See the Concept Check in this section. $$ 3 z^{2}=10 $$
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