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Chapter 9: Problem 15

Write in standard form. Use the quadratic formula to solve the equation. $$4 x^{2}+4 x=-1$$

Short Answer

Expert verified
The solution to the equation \(4 x^{2}+4 x=-1\) is \(x = -0.5\).

Step by step solution

01

Rearrange the equation in standard form

First, rewrite the equation \(4 x^{2}+4 x=-1\) as \(4 x^{2}+4 x+1=0\) which is now in the standard form of \(ax^2+bx+c=0\) where \(a = 4, b = 4, and c = 1\).
02

Apply the Quadratic Formula

Using the values of a, b, and c from the standard equation, plug these into the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). So, x equates to \(x = \frac{-4 \pm \sqrt{4^{2} - 4*4*1}}{2*4}\)
03

Simplify the roots

Solving inside of the square root first, \(4^{2} - 4*4*1 = 16 - 16 = 0\). Now the formula simplifies to \(x = \frac{-4 \pm \sqrt{0}}{8}\). Therefore, \(x = -0.5\) as the square root of 0 is 0 and -4 divided by 8 equals -0.5. We have only one real solution as the discriminant (b^2- 4ac) is equal to 0.

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Key Concepts

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

Standard Form
To tackle quadratic equations effectively, it's crucial to understand their structure in the standard form, which is given by the equation: \[ ax^2 + bx + c = 0 \]. Here, the coefficients are expressed as:
  • \( a \): The coefficient of \( x^2 \), which determines the parabola's direction (upwards if positive, downwards if negative).
  • \( b \): The coefficient of \( x \), which affects the symmetry of the parabola.
  • \( c \): The constant term, which shifts the parabola up or down on the graph.
Rewriting a quadratic equation in standard form helps set the stage for using other methods like factoring or the quadratic formula. For example, the equation \( 4x^2 + 4x = -1 \) was rewritten as \( 4x^2 + 4x + 1 = 0 \). This step is essential to define each coefficient clearly for future calculations.
Quadratic Formula
The quadratic formula is a powerful tool used to solve any quadratic equation, especially when factoring is not feasible. Its general form is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \],where:
  • \( b \): The coefficient of the linear term.
  • \( a \) and \( c \): Coefficients that were identified after arranging the equation in standard form.
Once the coefficients are known, substitute them into the formula to find the roots. For the quadratic \( 4x^2 + 4x + 1 = 0 \), let's calculate:- \( a = 4 \), \( b = 4 \), \( c = 1 \).- Plug these values into the quadratic formula: \[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 4 \cdot 1}}{2 \cdot 4} \].This method works systematically, ensuring that all possible roots, including irrational or complex numbers, are discovered.
Discriminant
The discriminant is a part of the quadratic formula under the square root, represented as \( b^2 - 4ac \). It determines the nature and number of roots the quadratic equation possesses:
  • If the discriminant is positive, it indicates two distinct real roots.
  • If it equals zero, like in the problem \( 4^2 - 4 \cdot 4 \cdot 1 = 0 \), there is exactly one real root.
  • If negative, the roots are complex, implying no real solutions.
In our problem, a zero discriminant led to one real solution \( x = -0.5 \). Understanding how the discriminant works helps predict the equation鈥檚 roots even before solving them. This can inform deeper insights into the nature of the solution, streamlining problem-solving strategies efficiently.

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Most popular questions from this chapter

Use the following information. Scientists simulate a gravity-free environment called microgravity in free- fall situations. A similar microgravity environment can be felt on free-fall rides at amusement parks or when stepping off a high diving platform. The distance \(d\) (in meters) that an object that is dropped falls in \(t\) seconds can be modeled by the equation \(d=\frac{1}{2} g\left(t^{2}\right),\) where \(g\) is the acceleration due to gravity (9.8 meters per second per second). If you want to double the free-fall time, how much do you have to increase the height from which the object was dropped?

In parts (a)-(d), a batter hits a pitched baseball when it is 3 feet off the ground. After it is hit, the height \(h\) (in feet) of the ball at time \(t\) (in seconds) is modeled by$$h=-16 t^{2}+80 t+3$$where \(t\) is the time (inseconds). a.Find the time when the ball hits the ground in the outfield. b.Write a quadratic equation that you can use to find the time when the baseball is at its maximum height of 103 feet. Solve the quadratic equation. c.Use a graphing calculator to graph the function. Use the zoom feature to approximate the time when the baseball is at its maximum height. Compare your results with those you obtained in part (b). d.What factors change the path of a baseball? What factors would contribute to hitting a home run?

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You see a firefighter aim a fire hose from 4 feet above the ground at a window that is 26 feet above the ground. The equation \(h=-0.01 d^{2}+1.06 d+4\) models the path of the water when \(h\) equals height in feet. Estimate, to the nearest whole number, the possible horizontal distances \(d\) (in feet) between the firefighter and the building.
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