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Chapter 2: Problem 48

Solve the quadratic equation by using the quadratic formula. Find only real solutions. $$-3 x^{2}+2 x-1=0$$

Short Answer

Expert verified
The quadratic equation \(-3x^2 + 2x - 1 = 0\) has no real solutions as the discriminant is negative.

Step by step solution

01

Identify a, b, and c from the quadratic equation

In a quadratic equation of the form \(ax^2 + bx + c = 0\), comparing it with the given equation \(-3x^2 + 2x - 1 = 0\), it can be seen that \(a = -3\), \(b = 2\), and \(c = -1\).
02

Compute the discriminant

The discriminant is calculated as \(b^2 - 4ac\). Substituting in the values, we get \(Discriminant = (2)^2 - 4*(-3)*(-1) = 4 - 12 = -8\).
03

Check the value of the discriminant

As the Discriminant is negative (-8), it implies that the given quadratic equation has no real solutions as square root of a negative number is not a real number.

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Key Concepts

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

Understanding Quadratic Equations
A quadratic equation is a type of polynomial equation of degree 2, typically in the form of \(ax^2 + bx + c = 0\). These equations are called 'quadratic' because "quadratus" is the Latin word for 'square', relating to the square of the variable \( x^2 \). The general setup consists of:
  • \( a \): the coefficient of \( x^2 \), which determines the parabola's openness and direction.
  • \( b \): the coefficient of \( x \), affecting the parabola's symmetry axis.
  • \( c \): the constant term, giving the vertical shift from the origin.
By comparing the given quadratic equation, \(-3x^2 + 2x - 1 = 0\), we identify \(a = -3\), \(b = 2\), and \(c = -1\). Solving quadratic equations often involves methods like factoring, completing the square, or using the quadratic formula, where the quadratic formula is useful when others are not readily applicable.
Role of the Discriminant
The discriminant is a crucial part of the quadratic formula, denoted as \(b^2 - 4ac\). It tells you about the number and types of solutions a quadratic equation may have. Specifically, it provides insight into whether solutions are real or imaginary. For any given quadratic equation, you compute the discriminant using \(a\), \(b\), and \(c\) values.
  • If \(b^2 - 4ac > 0\), there are two distinct real solutions.
  • If \(b^2 - 4ac = 0\), there's exactly one real solution (a repeated root).
  • If \(b^2 - 4ac < 0\), there are no real solutions, only two complex (imaginary) ones.
The provided equation \(-3x^2 + 2x - 1 = 0\) results in a discriminant of \(-8\), which is less than zero, indicating there are no real solutions for this equation.
Real Solutions in Quadratic Equations
Real solutions are values of \(x\) that satisfy the quadratic equation and form real numbers on the number line. They are derived from solving the quadratic equation using methods like the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Real solutions occur as pairs or single repeated roots:
  • Two real and distinct solutions arise when the discriminant is positive.
  • One real repeated solution exists when the discriminant is zero.
  • No real solutions are present when the discriminant is negative, as the results involve the square root of a negative number, creating complex numbers.
In the problem given, after calculating the discriminant as negative (-8), it confirms that our quadratic equation \(-3x^2 + 2x - 1 = 0\) has no real number solutions.

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Most popular questions from this chapter

Examine the following table of values for a quadratic function \(f\) $$\begin{array}{rr} x & f(x) \\ -2 & 3 \\ -1 & 0 \\ 0 & -1 \\ 1 & 0 \\ 2 & 3 \end{array}$$ (a) What is the equation of the axis of symmetry of the associated parabola? Justify your answer. (b) Find the minimum or maximum value of the function and the value of \(x\) at which it occurs. (c) Sketch a graph of the function from the values given in the table, and find an expression for the function.

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.

When using transformations with both vertical scaling and vertical shifts, the order in which you perform the transformations matters. Let \(f(x)=|x|\). (a) Find the function \(g(x)\) whose graph is obtained by first vertically stretching \(f(x)\) by a factor of 2 and then shifting the result upward by 3 units. A table of values and/or a sketch of the graph will be helpful. (b) Find the function \(g(x)\) whose graph is obtained by first shifting \(f(x)\) upward by 3 units and then multiplying the result by a factor of 2 A table of values and/or a sketch of the graph will be helpful. (c) Compare your answers to parts (a) and (b). Explain why they are different.

In Exercises \(105-110\), find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) \(h \neq 0,\) for the given function \(f\). $$f(x)=\frac{1}{x-3}, x \neq 3$$

A rectangular sandbox is to be enclosed with a fence on three of its sides and a brick wall on the fourth side. If 24 feet of fencing material is available, what dimensions will yield an enclosed region with an area of 70 square feet?
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