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Chapter 2: Problem 13

Solve the quadratic equation by factoring. $$-3 x^{2}+12=0$$

Short Answer

Expert verified
The solutions to the quadratic equation are \(x = 2\) and \(x = -2\).

Step by step solution

01

Rewrite Equation

First, rewrite the equation as \(-3 x^{2} + 12 = 0\) to \(3x^{2} - 12 = 0\). Here, both terms were divided by -1 to simplify the equation.
02

Factor the equation

Factor the equation into two terms. \(3x^{2} - 12 = 0\) is equivalent to \(3(x^{2} - 4) = 0\). Now factor the difference of squares inside the parenthesis to get \((x - 2)(x + 2)\). Multiply it back to the outside term 3 to get the final factored form as \(3(x - 2)(x + 2) = 0\)
03

Solve for x

Now, set each factor equal to zero, and then solve for x. This will give the solutions to the equation. For the factor \(x - 2\), the equation is \(x - 2 = 0\), and solving for x gives \(x = 2\). For the factor \(x + 2\), the equation is \(x + 2 = 0\), and solving for x gives \(x = -2\). ${\rm So} x=2$ or $x=-2$ is the solution.

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Key Concepts

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

Factoring
Factoring is like finding a pair of glasses that helps you see numbers and expressions more clearly by breaking them down into simpler parts. When we factor a quadratic equation like \(-3x^2 + 12 = 0\), we look for a way to express it as a product of simpler expressions.
  • First, simplify the equation by eliminating the negative sign. This step involves changing \(-3x^2 + 12 = 0\) to \(3x^2 - 12 = 0\) by dividing everything by -1.
  • Next, observe if there's a common factor in the simplified equation terms. Here, both terms 3 and 12 are divisible by 3.
  • Factor out the common factor. So the equation becomes \(3(x^2 - 4) = 0\).
The question now transforms into an easier problem about finding two numbers (\(x - 2\) and \(x + 2\)) that when multiplied together, along with 3, will give zero. Factoring reduces a complex equation into bite-sized pieces making it easier to handle and solve.
Difference of Squares
The difference of squares is a special type of factoring where an expression is made of two perfect squares separated by a minus sign. It follows the pattern \(a^2 - b^2 = (a-b)(a+b)\).
In the context of our quadratic equation \(3(x^2 - 4) = 0\), the expression \(x^2 - 4\) is a difference of squares:
  • Recognize \(x^2 - 4\) as a difference of squares since \(x^2\) and \(4\) (which is \(2^2\)) are perfect squares.
  • Apply the difference of squares formula \( a^2 - b^2 = (a-b)(a+b) \), where \(a = x\) and \(b = 2\).
This means \((x^2 - 4) = (x - 2)(x + 2)\).
The difference of squares allows us to quickly and easily break down equations that might seem complex at first glance.Exploit this property to simplify and solve expressions effectively by recognizing this pattern.
Solving Equations
After factoring a quadratic equation successfully, solving it becomes straightforward. This involves finding the values of \( x \) that satisfy the equation.
Here's how it's done:
  • Set each factor equal to zero. From \(3(x - 2)(x + 2) = 0\), you focus on the expressions inside the parenthesis: \(x - 2\) and \(x + 2\).
  • For \(x - 2\), set it equal to zero: \(x - 2 = 0\). Solving for \(x\) gives \(x = 2\).
  • For \(x + 2\), set it equal to zero: \(x + 2 = 0\). Solving for \(x\) gives \(x = -2\).
Both values of \( x \) satisfy the equation, so \(x = 2\) and \(x = -2\) are the solutions.Solving quadratic equations via factoring involves splitting the problem into smaller pieces, solving each part, and reuniting their solutions.

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Most popular questions from this chapter

Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$g(t)=t^{2}+t+1$$

The point (-2,2) on the graph of \(f(x)=|x|\) has been shifted horizontally and vertically to the point (3,4) Identify the shifts and write a new function \(g(x)\) in terms of \(f(x)\).

A rectangular garden plot is to be enclosed with a fence on three of its sides and an existing wall on the fourth side. There is 45 feet of fencing material available. (a) Write an equation relating the amount of available fencing material to the lengths of the three sides that are to be fenced. (b) Use the equation in part (a) to write an expression for the width of the enclosed region in terms of its length. (c) For each value of the length given in the following table of possible dimensions for the garden plot, fill in the value of the corresponding width. Use your expression from part (b) and compute the resulting area. What do you observe about the area of the enclosed region as the dimensions of the garden plot are varied? $$\begin{array}{ccc}\hline \begin{array}{c}\text { Length } \\\\\text { (feet) }\end{array} & \begin{array}{c}\text { Width } \\\\\text { (feet) }\end{array} & \begin{array}{c}\text { Total Amount of } \\\\\text { Fencing Material (feet) }\end{array} & \begin{array}{c}\text { Area } \\\\\text { (square feet) }\end{array} \\\\\hline 5 & & 45 \\\10 & & 45 \\\15 & & 45 \\\20 & & 45 \\\30 & & 45 \\\k & & 45 \\\& &\end{array}$$ (d) Write an expression for the area of the garden plot in terms of its length. (e) Find the dimensions that will yield a garden plot with an area of 145 square feet.

The shape of the Gateway Arch in St. Louis, Missouri, can be approximated by a parabola. (The actual shape will be discussed in an exercise in the chapter on exponential functions.) The highest point of the arch is approximately 625 feet above the ground, and the arch has a (horizontal) span of approximately 600 feet. (Check your book to see image) (a) Set up a coordinate system with the origin at the midpoint of the base of the arch. What are the \(x\) -intercepts of the parabola, and what do they represent? (b) What do the \(x\)- and \(y\)-coordinates of the vertex of the parabola represent? (c) Write an expression for the quadratic function associated with the parabola. (Hint: Use the vertex form of a quadratic function and find the coefficient \(a .)\) (d) What is the \(y\)-coordinate of a point on the arch whose (horizontal) distance from the axis of symmetry of the parabola is 100 feet?

When designing buildings, engineers must pay careful attention to how different factors affect the load a structure can bear. The following table gives the load in terms of the weight of concrete that can be borne when threaded rod anchors of various diameters are used to form joints. $$\begin{array}{cc} \text { Diameter (in.) } & \text { Load (1b) } \\ \hline 0.3750 & 2105 \\ 0.5000 & 3750 \\ 0.6250 & 5875 \\ 0.7500 & 8460 \\ 0.8750 & 11,500 \end{array}$$ (a) Examine the table and explain why the relationship between the diameter and the load is not linear. (b) The function $$f(x)=14,926 x^{2}+148 x-51$$ gives the load (in pounds of concrete) that can be borne when rod anchors of diameter \(x\) (in inches) are employed. Use this function to determine the load for an anchor with a diameter of 0.8 inch. (c) since the rods are drilled into the concrete, the manufacturer's specifications sheet gives the load in terms of the diameter of the drill bit. This diameter is always 0.125 inch larger than the diameter of the anchor. Write the function in part (b) in terms of the diameter of the drill bit. The loads for the drill bits will be the same as the loads for the corresponding anchors. (Hint: Examine the table of values and see if you can present the table in terms of the diameter of the drill bit.)
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