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Chapter 2: Problem 3

Find the \(x\) -intercepts of the given function. $$y=2 x^{2}+5 x+2$$

Short Answer

Expert verified
The x-intercepts are \[x = -\frac{1}{2}, -2\]

Step by step solution

01

Set up the equation

To find the x-intercepts, set the function equal to zero: \[2x^2 + 5x + 2 = 0\]
02

Factor the quadratic equation

Look for factors of the quadratic equation that multiply to the constant term (2) and add up to the coefficient of the middle term (5). After factoring, the equation becomes: \[(2x + 1)(x + 2) = 0\]
03

Solve for x

Set each factor equal to zero and solve for x: \[2x + 1 = 0\] Solve for x: \[x = -\frac{1}{2}\] For the other factor: \[x + 2 = 0\] Solve for x: \[x = -2\]

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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 the form \( ax^2 + bx + c = 0 \). The highest degree of the variable \(x\) is 2, which means that \(x\) is squared. Quadratic equations are very common in algebra and they generally represent parabolic shapes when graphed.
To encapsulate a clear understanding:
  • \(a, b,\) and \(c\) are constants.
  • The equation can open upwards or downwards depending on the sign of \(a\).
Quadratic equations model many real-life scenarios, such as projectile motion and areas of spaces.
Factoring
Factoring is a method used to break down polynomials into simpler, multiplied factors that when expanded give back the original polynomial. In the context of quadratic equations, factoring helps in finding the roots (solutions) of the equation.
When factoring the quadratic equation\(2x^2 + 5x + 2 = 0\):
  • Identify two numbers that multiply to \(ac\) (where \(a = 2\) and \(c = 2\), thus \(ac = 4\)), and add up to \(b\) (which is 5).
  • In this case, the numbers that satisfy these conditions are 4 and 1.
  • Rewrite the middle term with these numbers: \(2x^2 + 4x + x + 2\).
  • Factor by grouping: \((2x^2 + 4x) + (x + 2) = 2x(x + 2) + 1(x + 2) = (2x + 1)(x + 2)\).
You've now factored the quadratic equation into simpler terms.
Roots of Equation
The roots of an equation are the values of the variable that make the equation equal to zero. For a quadratic equation \( ax^2 + bx + c = 0 \), the solutions are found by setting the factored form to zero and solving for the variable.
Using our factored equation \( (2x + 1)(x + 2) = 0 \):
  • Set each factor equal to zero: \(2x + 1 = 0\) and \(x + 2 = 0\).
  • Solving the first: \(2x + 1 = 0\) gives \(x = -\frac{1}{2}\).
  • Solving the second: \(x + 2 = 0\) gives \(x = -2\).
Thus, the x-intercepts (roots) of the function \( y = 2x^2 + 5x + 2 \) are \( x = -\frac{1}{2} \) and \( x = -2 \).

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Most popular questions from this chapter

If \(f(x)\) is defined on the interval \(0 \leq x \leq 5\) and \(f^{\prime}(x)\) is negative for all \(x,\) for what value of \(x\) will \(f(x)\) have its greatest value? Explain why.

The Great American Tire Co. expects to sell 600,000 tires of a particular size and grade during the next year. Sales tend to be roughly the same from month to month. Setting up each production run costs the company \(\$ 15,000 .\) Carrying costs, based on the average number of tires in storage, amount to \(\$ 5\) per year for one tire. (a) Determine the costs incurred if there are 10 production runs during the year. (b) Find the economic lot size (that is, the production run size that minimizes the overall cost of producing the tires).

Advertising for a certain product is terminated, and \(t\) weeks later, the weekly sales are \(f(t)\) cases, where $$f(t)=1000(t+8)^{-1}-4000(t+8)^{-2}.$$ At what time is the weekly sales amount falling the fastest?

A certain airline requires that rectangular packages carried on an airplane by passengers be such that the sum of the three dimensions (i.e., length, width, and height), is at most 120 centimeters. Find the dimensions of the square ended rectangular package of greatest volume that meets this requirement.

The monthly demand equation for an electric utility company is estimated to be $$p=60-\left(10^{-5}\right) x,$$ where \(p\) is measured in dollars and \(x\) is measured in thousands of kilowatt- hours. The utility has fixed costs of 7 million dollars per month and variable costs of \(\$ 30\) per 1000 kilowatt-hours of electricity generated, so the cost function is $$a(x)=7 \cdot 10^{6}+30 x,$$ (a) Find the value of \(x\) and the corresponding price for 1000 kilowatt-hours that maximize the utility's profit. (b) Suppose that rising fuel costs increase the utility's variable costs from \(\$ 30\) to \(\$ 40\), so its new cost function is $$C_{1}(x)=7 \cdot 10^{6}+40 x.$$ Should the utility pass all this increase of \(\$ 10\) per thousand kilowatt- hours on to consumers? Explain your answer.
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