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Chapter 3: Problem 37

Solve each quadratic equation (a) graphically, (b) numerically, and (c) symbolically. Express graphical and numerical solutions to the nearest tenth when appropriate. $$ x^{2}-x-6=0 $$

Short Answer

Expert verified
The solutions are \(x = 3\) and \(x = -2\).

Step by step solution

01

Understanding the Problem

We need to find the values of \(x\) that satisfy the quadratic equation \(x^2 - x - 6 = 0\). This can be done graphically, numerically, and symbolically.
02

Solve Graphically

To solve graphically, we plot the equation \(y = x^2 - x - 6\) and find the x-values where the graph intersects the x-axis. Plot the points for x-values from, say, -3 to 5, and observe the curve. Using a graphing tool, the curve crosses the x-axis approximately at \(x = -2\) and \(x = 3\).
03

Solve Numerically

Numerically solve using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]For the equation \(x^2 - x - 6 = 0\), let \(a = 1\), \(b = -1\), and \(c = -6\). Substitute these into the formula:\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \times 1 \times (-6)}}{2 \times 1} \]\[ x = \frac{1 \pm \sqrt{1 + 24}}{2} \]\[ x = \frac{1 \pm \sqrt{25}}{2} \]\[ x = \frac{1 \pm 5}{2} \]This gives us the numerical solutions \(x = 3\) and \(x = -2\).
04

Solve Symbolically

Factor the equation \(x^2 - x - 6 = 0\) by finding two numbers that multiply to -6 and add to -1. These numbers are -3 and 2. So, factor the equation:\[(x - 3)(x + 2) = 0\]Set each factor equal to zero:\[x - 3 = 0\] or \[x + 2 = 0\]Solve these to get \(x = 3\) and \(x = -2\).
05

Confirm Solutions

Check each solution in the original equation. For \(x = 3\): \[3^2 - 3 - 6 = 0\]: checks out. For \(x = -2\):\[(-2)^2 + 2 - 6 = 0\]: checks out. Both solutions are correct.

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Key Concepts

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

Graphical Solutions
Graphical solutions involve plotting the quadratic equation on a graph to identify where the curve crosses the x-axis. For the equation \( y = x^2 - x - 6 \), graphing the function allows us to visualize the solutions. The points where the curve intersects the x-axis are the solutions to the equation. In this case, the curve crosses the x-axis at approximately \( x = -2 \) and \( x = 3 \).
Using a graph provides an intuitive understanding of the roots' approximate values by visually inspecting the curve.
This method is particularly helpful when an exact answer isn't necessary or when the equation doesn't factor easily.
Numerical Methods
Numerical methods, such as using the quadratic formula, provide precise solutions for quadratic equations. Given the equation \( x^2 - x - 6 = 0 \), we use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -1 \), and \( c = -6 \). After substituting these values, we find:
\[x = \frac{1 \pm \sqrt{25}}{2} \]
This results in the solutions \( x = 3 \) and \( x = -2 \).
Numerical methods help in finding exact solutions, and they are especially beneficial when factoring isn't straightforward or possible.
Symbolic Solutions
Symbolic solutions focus on manipulating the equation algebraically to find exact solutions. For \( x^2 - x - 6 = 0 \), we attempt to factor it into simpler expressions.
To factor, look for two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\).
The equation factors to \((x - 3)(x + 2) = 0\).
Setting each factor equal to zero gives us the solutions \( x = 3 \) and \( x = -2 \).
Symbolic solutions, like factoring, offer a straightforward way to solve quadratics when possible, providing insight into the equation's structure.
Quadratic Formula
The quadratic formula is a powerful algebraic tool applicable to any quadratic equation. It finds the roots of quadratics by using the coefficients of the terms. The formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula is derived from the process of completing the square on the general quadratic equation \( ax^2 + bx + c = 0 \).
One key advantage is that it works universally, even with equations that don't factor easily.
This makes it indispensable for solving all types of quadratic equations, ensuring reliable solutions across varied problems.

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Most popular questions from this chapter

A cylindrical aluminum can is being constructed to have a height \(h\) of 4 inches. If the can is to have a volume of 28 cubic inches, approximate its radius \(r .\) (Hint: \(V=\pi r^{2} h\).)

As the altitude increases, air becomes thinner, or less dense. An approximation of the density of air at an altitude of \(x\) meters above sea level is given by \(d(x)=\left(3.32 \times 10^{-9}\right) x^{2}-\left(1.14 \times 10^{-4}\right) x+1.22\) The output is the density of air in kilograms per cubic meter. The domain of \(d\) is \(0 \leq x \leq 10,000\). (a) Denver is sometimes referred to as the mile-high city. Compare the density of air at sea level and in Denver. (Hint: 1 ft \(\approx 0.305\) m.) (b) Determine the altitudes where the density is greater than 1 kilogram per cubic meter.

One airline ticket costs \(\$ 250\). For each additional airline ticket sold to a group, the price of every ticket is reduced by \(\$ 2 .\) For example, 2 tickets \(\operatorname{cost} 2 \cdot 248=\$ 496\) and 3 tickets \(\operatorname{cost} 3 \cdot 246=\$ 738\) (a) Write a quadratic function that gives the total cost of buying \(x\) tickets. (b) What is the cost of 5 tickets? (c) How many tickets were sold if the cost is \(\$ 5200 ?\) (d) What number of tickets sold gives the greatest cost?

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 2 x^{2}+3 x=12-2 x $$

Suppose that a person's heart rate, \(x\) minutes after vigorous exercise has stopped, can be modeled by \(f(x)=\frac{4}{5}(x-10)^{2}+80\). The output is in beats per minute, where the domain of \(f\) is \(0 \leq x \leq 10\). (a) Evaluate \(f(0)\) and \(f(2)\). Interpret the result. (b) Estimate the times when the person's heart rate was between 100 and 120 beats per minute, inclusive.
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