Problem 37
Question
Solve each quadratic equation (a) graphically, (b) numerically, and (c) symbolically. Express graphical and numerical solutions to the nearest tenth when appropriate. $$ x^{2}-x-6=0 $$
Step-by-Step Solution
Verified Answer
The solutions are \(x = 3\) and \(x = -2\).
1Step 1: 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.
2Step 2: 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 \(x = 3\).
3Step 3: 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 \(x = -2\).
4Step 4: 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 \(x = -2\).
5Step 5: 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.
Key Concepts
Graphical SolutionsNumerical MethodsSymbolic SolutionsQuadratic Formula
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.
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.
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
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.
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.
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
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.
Other exercises in this chapter
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