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Chapter 1: Problem 22

Solve each equation by the zero-factor property. $$9 x^{2}-12 x+4=0$$

Short Answer

Expert verified
The solution to the equation is \(x = \frac{2}{3}\).

Step by step solution

01

Identify the form of the equation

The equation is in the form of a quadratic equation: \(ax^2 + bx + c = 0\), where a, b, and c are constants. In this case, \(a = 9\), \(b = -12\), and \(c = 4\).
02

Express the quadratic equation in factored form

To solve using the zero-factor property, we need to factor the quadratic equation. Notice that the left side of the equation resembles a perfect square trinomial:\(9x^2 - 12x + 4 = (3x - 2)^2\).
03

Set each factor to zero

Using the factored form \((3x - 2)^2 = 0\), apply the zero-factor property which states that if \(ab=0\) then either \(a=0\) or \(b=0\).Since it is a perfect square, we set \(3x - 2=0\).
04

Solve for x

Solve the equation \(3x - 2 = 0\). First, isolate x: Add 2 to both sides: \(3x = 2\) Now, divide by 3:\(x = \frac{2}{3}\).As it's a square term \((3x - 2)^2\) it does not give multiple solutions.

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Key Concepts

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

quadratic equation
A quadratic equation is a type of polynomial equation of degree two. The general form is: \[ ax^2 + bx + c = 0 \] where the coefficients \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable.
A quadratic equation is characterized by having exactly one squared term (\( x^2 \)).
When solving quadratic equations, several methods can be used:
  • Factoring
  • Using the quadratic formula
  • Completing the square
  • Graphing
Depending on the equation, some methods might be easier than others. Now, let鈥檚 explore the factored form, which is often our first go-to when we encounter quadratic equations.
factored form
To solve a quadratic equation using the zero-factor property, we first need to express the quadratic equation in factored form. The factored form of a quadratic equation is: \[ (px + q)(rx + s) = 0 \] In this form, setting each factor equal to zero allows us to solve for \(x\).

For example:
The quadratic equation \(9x^2 - 12x + 4 = 0\) can be factored into:
\[ (3x - 2)^2 = 0 \]
Here, \( (3x - 2) \) is a repeated factor, known as a perfect square trinomial.
Applying the zero-factor property involves setting each factor to zero: \[ 3x - 2 = 0 \] By solving this simple linear equation, we can isolate \( x \) and find its value.
perfect square trinomial
A perfect square trinomial is a special form of a quadratic equation that looks like the square of a binomial. It takes the form:
  • \( (ax + b)^2 \)
  • \( (ax - b)^2 \)
For example:
\( 9x^2 - 12x + 4 \) can be written as:
\[ (3x - 2)^2 \] Here, \( (3x - 2) \) is the binomial.
A perfect square trinomial makes solving quadratic equations easier because the zero-factor property can be applied directly.
When you see \( (px + q)^2 = 0 \), simply set the inside factor equal to zero: \[ px + q = 0 \]
By solving for \( x \), you find the solution to the equation: \[ x = -\frac{q}{p} \] This straightforward process helps unravel complex-looking quadratic equations with ease.

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Most popular questions from this chapter

Solve each rational inequality. Write each solution set in interval notation. \(4\frac{x+2}{3+2 x} \leq 5\)4

Find each product. Write the answer in standard form. $$(2+i)(2-i)(4+3 i)$$

Simplify each power of i. $$i^{23}$$

If p units of an item are sold for \(x\) dollars per unit, the revenue is \(R=p x\). Use this idea to analyze the following problem. Number of Apartments Rented The manager of an 80-unit apartment complex knows from experience that at a rent of \(\$ 300,\) all the units will be full. On the average, one additional unit will remain vacant for each \(\$ 20\) increase in rent over \(\$ 300 .\) Furthermore, the manager must keep at least 30 units rented due to other financial considerations. Currently, the revenue from the complex is \(\$ 35,000 .\) How many apartments are rented? Suppose that \(x\) represents the number of \(\$ 20\) increases over \(\$ 300 .\) Represent the number of apartment units that will be rented in terms of \(x .\)

Solve each equation or inequality. $$|5 x+2|-2<3$$
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