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Chapter 0: Problem 60

Solve the quadratic equation by factoring $$ 4 x^{2}+12 x+9=0 $$

Short Answer

Expert verified
The root of the quadratic equation \(4x^2 + 12x + 9 = 0\) is x = -1.5.

Step by step solution

01

Identify the values of a, b, and c

The given equation is \( 4x^2 + 12x + 9 = 0 \), it's clear that a = 4, b = 12, and c = 9.
02

Write the equation in factored form

The factored form of the equation \(4x^2 + 12x + 9 = 0\) is \(4(x + 1.5)^2 = 0\). This achieved by completing the square method where \(x^2 + bx + (b/2a)^2 = a(x + b/2a)^2\).
03

Set each factor equal to zero and solve for x

(x + 1.5)^2 = 0. Then x = -1.5. Since we only have one factor, there is only one root (x = -1.5).

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Key Concepts

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

Factoring Method
The factoring method is a common and practical technique to solve quadratic equations. It involves rewriting the quadratic equation so it can be expressed as a product of two binomials. This means taking the quadratic equation in the form of \(ax^2 + bx + c = 0\) and finding two expressions that multiply together to give you the original quadratic.In the equation \(4x^2 + 12x + 9 = 0\), the coefficients \(a = 4\), \(b = 12\), and \(c = 9\) play a key role in determining the needed factors.
  • When factoring, it's important to look for two numbers that multiply to give \(ac\) (product of \(a\) and \(c\)) and add up to \(b\).
  • However, in the case of perfect square trinomials like \(4x^2 + 12x + 9\), it can be factored to \((2x + 3)^2\).
This means we can rewrite the equation as \((2x + 3)(2x + 3) = 0\) or simplified to \((2x + 3)^2 = 0\). This directly leads us to determine the roots easily once factored.
Completing the Square
Completing the square is another method to solve quadratic equations by transforming the equation into a perfect square trinomial. This technique can give you deeper insights into the properties of the quadratic equation, like its vertex.To complete the square for \(4x^2 + 12x + 9\):
  • Start by dividing every term by \(a = 4\) to simplify the process: \(x^2 + 3x + 2.25\).
  • Then, form a perfect square trinomial by adding and subtracting the square of half the coefficient of \(x\). Here, half of \(3\) is \(1.5\), and its square is \(2.25\).
  • This rearranges the equation into \((x + 1.5)^2 = 0\).
The whole idea is to make the quadratic part of the equation look like a square of a binomial, which allows you to solve for \(x\) by taking the square root of both sides.
Roots of a Quadratic Equation
Finding the roots of a quadratic equation is about discovering the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These are often referred to as "solutions" or "zeros" of the equation.Once you have rewritten the quadratic equation into its factored form or completed the square, finding the roots becomes straightforward:
  • If the equation is factored into \((x - r_1)(x - r_2) = 0\), then \(x = r_1\) and \(x = r_2\) are the roots.
  • In this exercise, the quadratic \(4x^2 + 12x + 9\) was transformed into \((x+1.5)^2 = 0\).
This indicates a repeated root, with only one unique solution \(x = -1.5\). A repeated root occurs when the discriminant of the quadratic equation (\(b^2 - 4ac\)) equals zero, resulting in a perfect square and hence only one solution.

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Most popular questions from this chapter

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)= \begin{cases}x+3, & x<0 \\ 6-x, & x \geq 0\end{cases} $$

True or False? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. $$ \text { If } f \text { is an even function, } f^{-1} \text { exists. } $$

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ g(x)=\frac{x}{8} $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=3 x+1 $$

Average Price The average prices \(p\) (in thousands of dollars) of a new mobile home in the United States from 1990 to 2002 (see figure) can be approximated by the model $$ p(t)= \begin{cases}0.182 t^{2}+0.57 t+27.3, & 0 \leq t \leq 7 \\ 2.50 t+21.3, & 8 \leq t \leq 12\end{cases} $$ where \(t\) represents the year, with \(t=0\) corresponding to 1990. Use this model to find the average price of a mobile home in each year from 1990 to 2002 . (Source: U.S. Census Bureau)
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