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Chapter 2: Problem 5

Write the quadratic equation in general form. Do not solve the equation. $$2 x^{2}=3-5 x$$

Short Answer

Expert verified
The quadratic equation in general form is \(2x^{2} + 5x - 3 = 0\).

Step by step solution

01

Understand the general form

The general form of a quadratic equation is \(ax^{2} + bx + c = 0\). Here, 'a' is the coefficient of \(x^2\), 'b' is the coefficient of 'x', and 'c' is the constant term.
02

Rearrange the equation

Rearrange the given equation \(2x^{2} = 3 - 5x\) into the general form. To do this, add \(5x\) to both sides to shift the \(x\) term to the left and obtain: \(2x^{2} + 5x = 3\).
03

Shift the constant to the left

To complete the transformation into the general form, move the constant to the left of the equation by subtracting 3 from both sides. This gives the final equation: \(2x^{2} + 5x - 3 = 0\).

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Key Concepts

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

General Form
Understanding the general form of a quadratic equation is essential when working with quadratic equations. The general form is a standard way of writing any quadratic equation, expressed as: - \( ax^2 + bx + c = 0 \).
  • Here, 'a' is the coefficient that multiplies with the squared term \(x^2\).
  • 'b' is the coefficient of the first-degree term, x.
  • 'c' is the constant term without any variable attached to it.
The general form is used because it neatly organizes the equation components, making it easier to apply methods for solving such equations. When a quadratic equation is in this form, it's straightforward to identify the coefficients and constant, which are critical for solving the equation through various methods like the quadratic formula or factorization.
Coefficients
In mathematics, coefficients are numerical or constant factors in terms of an algebraic expression. For quadratic equations, understanding coefficients is crucial:
  • The coefficient 'a' is particularly important because it affects the parabola's direction of opening. If \(a > 0\), the parabola opens upwards, whereas if \(a < 0\), it opens downwards. This greatly influences the shape and nature of the graph.
  • The coefficient 'b' combines with 'a' and 'c' in determining the vertex of the parabola, as well as the axis of symmetry.
For example, in the quadratic equation obtained from the exercise, \(2x^2 + 5x - 3 = 0\), the coefficients are 2 for \(x^2\) and 5 for \(x\). These coefficients together influence many of the properties of the quadratic, from its roots to its graphical representation.
Constants
Constants in algebra, especially in quadratic equations, denote fixed values that do not change with x. In a quadratic equation's general form, \( ax^2 + bx + c = 0 \), 'c' represents the constant term. This constant has significant effects:
  • It shifts the entire graph of the equation vertically on a coordinate plane. This means if you change 'c', it won't change the shape of the graph but will move it up or down.
  • In calculating the roots of the equation using methods like the quadratic formula, 'c' plays a role in determining the sum and product of the roots.
In the exercise, the constant was identified as -3 after rearranging the terms into general form. Its value influences both the equation's graphical representation and its algebraic solutions.

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Most popular questions from this chapter

use the models below, which approximate the numbers of Bed Bath \(\&\) Beyond stores \(B\) and Williams-Sonoma stores \(W\) for the years 2000 through \(2013,\) where \(t\) is the year, with \(t=0\) corresponding to 2000. (Sources: Bed Bath \(\&\) Beyond, Inc.; Williams-Sonoma, Inc.) Bed Bath \& Beyond: $$B=86.5 t+342,0 \leq t \leq 13$$ Williams-Sonoma: $$W=-2.92 t^{2}+52.0 t+381,0 \leq t \leq 13$$. Solve the inequality \(B(t) \geq 900 .\) Explain what the solution of the inequality represents.

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=3 x-3 \sqrt{x}-4$$

Use a graphing utility to graph the equation and graphically approximate the values of \(x\) that satisfy the specified inequalities. Then solve each inequality algebraically. Equation \(y=-x^{2}+2 x+3\) Inequalities (a) \(y \leq 0\) (b) \(y \geq 3\)

Given that the solutions of a quadratic equation are \(x=(-b \pm \sqrt{b^{2}-4 a c}) /(2 a),\) show that the product of the solutions is \(P=c / a\).

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$6-\frac{1}{x}-\frac{1}{x^{2}}=0$$
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