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Chapter 2: Problem 50

Find the solutions of the inequality by drawing appropriate graphs. State each answer correct to two decimals. $$ 0.5 x^{2}+0.875 x \leq 0.25 $$

Short Answer

Expert verified
The solution is \(-2.00 \leq x \leq 0.25\).

Step by step solution

01

Rewrite the Inequality as an Equation

First, we treat the inequality as an equation. Rewrite it as:\[ 0.5x^2 + 0.875x = 0.25 \]We'll analyze this equation to understand its graph.
02

Move All Terms to One Side

Subtract 0.25 from both sides of the equation:\[ 0.5x^2 + 0.875x - 0.25 = 0 \]This will help us factor or use the quadratic formula to solve for roots.
03

Use the Quadratic Formula

Apply the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \( a = 0.5 \), \( b = 0.875 \), and \( c = -0.25 \).
04

Calculate the Discriminant

Compute the discriminant \( b^2 - 4ac \):\[ (0.875)^2 - 4 \cdot 0.5 \cdot (-0.25) \]\[ 0.765625 + 0.5 = 1.265625 \]Since the discriminant is positive, the quadratic equation has two distinct real roots.
05

Solve for Roots

Using the quadratic formula:\[ x = \frac{-0.875 \pm \sqrt{1.265625}}{1} \]Calculate the roots:\[ \sqrt{1.265625} \approx 1.125 \]\[ x_1 = \frac{-0.875 + 1.125}{1} \approx 0.25 \]\[ x_2 = \frac{-0.875 - 1.125}{1} \approx -2.00 \]
06

Analyze the Parabola

The roots found, \( x_1 = 0.25 \) and \( x_2 = -2.00 \), are the points where the parabola intersects the x-axis. The parabola opens upwards (since the coefficient of \( x^2 \) is positive).
07

Determine the Solution Set

The inequality \( 0.5x^2 + 0.875x \leq 0.25 \) implies that the parabola is below or touching the x-axis. Thus, the solutions are between the roots:\[ -2.00 \leq x \leq 0.25 \]

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Key Concepts

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

Quadratic Equations
Quadratic equations are fundamental in algebra, usually in the form of \( ax^2 + bx + c = 0 \). They include terms up to the second degree, which means the highest power of \( x \) is squared. In this type of equation:
  • \( a \), \( b \), and \( c \) are constants, where \( a eq 0 \).
  • \( ax^2 \) determines the parabola's direction: upward if \( a > 0 \) and downward if \( a < 0 \).
Quadratic equations can be solved using various methods:
  • Factoring, if the equation can be split into two binomials.
  • Completing the square, to transform into a perfect square trinomial.
  • Quadratic formula, which is the most universal method:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula gives the values of \( x \) where the equation equals zero, known as the roots.
Parabola
The parabola is the graph of a quadratic equation. It is a U-shaped curve that can open upwards or downwards depending on the coefficient of the squared term:\( ax^2 \).
  • If \( a > 0 \), the parabola opens upwards.
  • If \( a < 0 \), the parabola opens downwards.
Key features of a parabola include:
  • Vertex: The highest or lowest point of the parabola.
  • Axis of symmetry: A vertical line through the vertex, dividing the parabola into two mirror images.
  • Roots/Intersections: Points where the parabola crosses the x-axis (solutions to the equation).
In our example, with \( ax^2 = 0.5x^2 \), the parabola opens upwards. The roots \( x_1 = 0.25 \) and \( x_2 = -2.00 \) are where the parabola intersects the x-axis.
Inequality Solving
Solving inequalities involves finding the set of values that satisfy the inequality. In quadratic inequalities, we deal with expressions like \( ax^2 + bx + c \leq 0 \). To solve them effectively:
  • Start by solving the corresponding equation \( ax^2 + bx + c = 0 \) to find its roots.
  • Draw the graph of the quadratic equation to visualize the inequality.
  • Understand where the curve is above or below the x-axis, based on the inequality's direction.
  • Identify intervals where the inequality holds. For \( \leq \) or \( \geq \), include the values where the curve just touches the x-axis.
In the exercise, we used the parabola's roots to find the interval \([ -2.00, 0.25 ]\), which indicates where the expression is less than or equal to zero.
Discriminant
The discriminant is key in understanding the nature of the roots of a quadratic equation. Calculated as \( b^2 - 4ac \), it shows:
  • If \( \text{Discriminant} > 0 \): Two distinct real roots.
  • If \( \text{Discriminant} = 0 \): One real root (both roots are the same, known as a repeated root).
  • If \( \text{Discriminant} < 0 \): No real roots, the roots are complex or imaginary.
In our problem, the discriminant calculated was 1.265625, which is positive. This tells us there are two distinct real roots for the equation. Consequently, these roots help us sketch the parabola and determine the interval of solutions for the inequality.

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