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Chapter 8: Problem 12

Use the discriminant to determine the number and type of solutions for each equation. Do not solve. \(6 x^{2}-5 x-6=0\)

Short Answer

Expert verified
The equation has two distinct real solutions.

Step by step solution

01

Identify coefficients

The general form of a quadratic equation is \(ax^2 + bx + c = 0\). For the given equation \(6x^2 - 5x - 6 = 0\), identify the coefficients: \(a = 6\), \(b = -5\), \(c = -6\).
02

Write the discriminant formula

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\). This formula helps to determine the number and types of solutions.
03

Substitute the coefficients into the formula

Substitute \(a = 6\), \(b = -5\), and \(c = -6\) into the discriminant formula: \(D = (-5)^2 - 4 imes 6 imes (-6)\).
04

Compute the discriminant

Calculate the discriminant: \(D = 25 - 4 imes 6 imes (-6) = 25 + 144 = 169\).
05

Determine the nature of solutions

Based on the discriminant value: if \(D > 0\), there are two distinct real solutions; if \(D = 0\), there is one real solution; and if \(D < 0\), there are two complex solutions. Here, \(D = 169\), which is greater than 0, indicating there are two distinct real solutions.

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Key Concepts

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

Understanding the Discriminant
The discriminant is a key concept in solving quadratic equations, often represented by the symbol \(D\). It provides valuable information about the nature of the solutions of a quadratic equation without having to solve it. For any quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant is calculated using the formula: \(D = b^2 - 4ac\).
Understanding its role is crucial:
  • **\(D > 0\)**: The equation has two distinct real solutions. This means the parabola intersects the x-axis at two points.
  • **\(D = 0\)**: There's exactly one real solution. Here, the parabola touches the x-axis at only one point, known as a repeated root.
  • **\(D < 0\)**: The solutions are complex and not real, meaning the parabola doesn't intersect the x-axis at any point.
The discriminant thus helps anticipate the type and number of solutions right from the equation itself.
Real Solutions of Quadratic Equations
When we talk about real solutions in the context of a quadratic equation, we're referring to the values of \(x\) that actually make the equation true in terms of real numbers. This is distinctly different from non-real (complex) solutions, which involve imaginary numbers.
Using the discriminant as a guide:
  • If \(D > 0\), two distinct real numbers satisfy the equation. This is often what we expect in typical real-world scenarios where inputs and outputs are measurable quantities.
  • When \(D = 0\), a single real number makes the equation true. This situation can also be interpreted as the equation having a repeated or double root.
Real solutions are vital for modeling and solving real-life problems, such as calculating time, distance, and other physical properties.
Identifying Coefficients in Quadratic Equations
Coefficients in a quadratic equation provide essential clues about the equation's shape and position. These are the numerical factors before variables in the standard form \(ax^2 + bx + c = 0\). In this format:
  • **\(a\)**: This is the leading coefficient, which affects the parabola's width and direction (upward or downward opening).
  • **\(b\)**: Known as the linear coefficient, it influences the parabola's axis of symmetry and its position along the x-axis.
  • **\(c\)**: This constant term affects the parabola's vertical position, determining where it crosses the y-axis.
Each of these coefficients plays a role in the discriminant formula \(D = b^2 - 4ac\), making it important to correctly identify and substitute them into the formula. Accurate computation of \(D\) helps predict the types of solutions the equation will have.

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Most popular questions from this chapter

Change each radical to an exponential expression. $$ \sqrt[7]{8 r^{2} s} $$

Solve each equation. Approximate the solutions to the nearest hundredth when appropriate. $$x^{2}-3 x-27=0$$

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Use a graphing calculator to solve each equation. If an answer is not exact, round to the nearest hundredth. See Using Your Calculator: Solving Quadratic Equations Graphically. Fireworks. \(\quad\) A fireworks shell is shot straight up with an initial velocity of 120 feet per second. Its height \(s\) in feet after \(t\) seconds is approximated by the equation \(s=120 t-16 t^{2} .\) If the shell is designed to explode when it reaches its maximum height, how long after being fired, and at what height, will the fireworks appear in the sky?

The vertex of a quadratic function \(f(x)=a x^{2}+b x+c\) is given by the formula \(\left(-\frac{b}{2 a}, f\left(-\frac{b}{2 a}\right)\right) .\) Explain what is meant by the notation \(f\left(-\frac{b}{2 a}\right)\)
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