Problem 78
Question
Use the intersect feature of your graphing calculator to explore the real solution(s), if any, of \(x^{2}=x+k\) for \(k=0, k=-\frac{1}{4},\) and \(k=-3 .\) Also use the zero feature to explore the solution(s). Relate your observations to the quadratic formula.
Step-by-Step Solution
Verified Answer
For \(k = 0\), the solutions are \(0\) and \(1\). For \(k = -1/4\), there is one solution at \(0.5\). For \(k = -3\), there are no real solutions. The solutions obtained are consistent across the three different methods of solving the quadratic equation, demonstrating their equivalence.
1Step 1: Solve the equation graphically
Use the graphing calculator to plot the equation \(x^{2}=x+k\). For \(k=0\), the equation is \(x^{2}=x\). When plotting on a graph, we see the quadratic \(x^{2}\) intersects the line \(x\) at two points where \(x = 0\) and \(x = 1\), giving us the solutions.Similarly, replace \(k\) with \(-1/4\) and \(-3\) and find the points of intersection.
2Step 2: Use the zero feature
The zero feature of the graphing calculator can be used to verify the solutions found visually. Set the graphed function equal to zero (\(x^{2}-x-k= 0\)) and use the 'Zero' feature to find the roots of the equations.
3Step 3 : Apply the quadratic formula
The quadratic formula, which solves equations of the form \(ax^{2} + bx + c = 0\), is \( x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \). In the given equation, \(a = 1\), \(b = -1\), and \(c = -k\). Substituting these values into the formula will yield the solutions for different values of \(k\).
4Step 4: Comparison of Methods
The real solutions obtained through the graph method, the 'Zero' feature, and the quadratic formula should be the same. This consistency is expected because the graphical method, the zero feature, and the quadratic formula are all different ways to solve the same quadratic equation.
Key Concepts
Quadratic FormulaZero Feature on Graphing CalculatorsGraphical Solution of Equations
Quadratic Formula
The quadratic formula is a critical tool for solving equations of the second degree, commonly known as quadratic equations. These equations are of the form \( ax^2 + bx + c = 0 \), where \( a \) is not zero. The general solution to these equations is given by \( x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}} \).
To apply the quadratic formula, you identify the coefficients \( a \) (the coefficient of \( x^2 \) term), \( b \) (the coefficient of \( x \) term), and \( c \) (the constant term) from your quadratic equation. Afterward, these values are substituted into the formula, providing potentially two solutions, accounting for the \( \pm \) sign. This reflects the fact that a parabola (the graph of a quadratic equation) can intersect the \( x \) axis at zero, one, or two points.
The quadratic formula is especially useful because it provides solutions to quadratic equations even when they do not factor simply and are not easily solved by other means. However, the formula requires careful operations with signs and square roots, so attention to detail is crucial.
To apply the quadratic formula, you identify the coefficients \( a \) (the coefficient of \( x^2 \) term), \( b \) (the coefficient of \( x \) term), and \( c \) (the constant term) from your quadratic equation. Afterward, these values are substituted into the formula, providing potentially two solutions, accounting for the \( \pm \) sign. This reflects the fact that a parabola (the graph of a quadratic equation) can intersect the \( x \) axis at zero, one, or two points.
The quadratic formula is especially useful because it provides solutions to quadratic equations even when they do not factor simply and are not easily solved by other means. However, the formula requires careful operations with signs and square roots, so attention to detail is crucial.
Zero Feature on Graphing Calculators
Modern graphing calculators have various features to aid in solving equations, one of which is the 'Zero' feature. This built-in tool allows students to easily find the roots of a quadratic equation—the points where the graph of the equation intersects the \( x \) axis, also known as zeros.
To use the zero feature, you first graph the equation in the form \( y = ax^2 + bx + c \). Then, the zero feature is activated which prompts you to select a point near the expected zero on the graph. The calculator then uses numerical methods to find the exact value of \( x \) that makes \( y \)= 0. This approach is particularly helpful in exploring the solutions for different values of parameters in an equation, like the variable \( k \) in the exercise provided.
The advantage of the 'Zero' feature is that it provides a visual representation of the solutions and aids in understanding the behavior of the quadratic function. It is also great for checking solutions derived algebraically, ensuring a comprehensive grasp of quadratic equations.
To use the zero feature, you first graph the equation in the form \( y = ax^2 + bx + c \). Then, the zero feature is activated which prompts you to select a point near the expected zero on the graph. The calculator then uses numerical methods to find the exact value of \( x \) that makes \( y \)= 0. This approach is particularly helpful in exploring the solutions for different values of parameters in an equation, like the variable \( k \) in the exercise provided.
The advantage of the 'Zero' feature is that it provides a visual representation of the solutions and aids in understanding the behavior of the quadratic function. It is also great for checking solutions derived algebraically, ensuring a comprehensive grasp of quadratic equations.
Graphical Solution of Equations
Graphical solutions of equations involve finding the intersection points of graphs that represent the equations. This visual approach is ideal for understanding the behavior of equations and the nature of their solutions.
For a quadratic equation like \( x^2 = x + k \), you plot two separate functions: \( y = x^2 \) and \( y = x + k \). The points where these two graphs intersect corresponds to the solutions of the equation. By varying the parameter \( k \) and observing the graph's changes, students can better understand how the position and the number of the intersections (solutions) are affected.
Graphical solutions are a great supplement to analytical methods like the quadratic formula or the zero feature on graphing calculators. By combining these approaches, students get both a conceptual and practical understanding. Also, graphical solutions help to visually interpret real-world situations modeled by quadratic equations, an essential skill in many areas of science and engineering.
For a quadratic equation like \( x^2 = x + k \), you plot two separate functions: \( y = x^2 \) and \( y = x + k \). The points where these two graphs intersect corresponds to the solutions of the equation. By varying the parameter \( k \) and observing the graph's changes, students can better understand how the position and the number of the intersections (solutions) are affected.
Graphical solutions are a great supplement to analytical methods like the quadratic formula or the zero feature on graphing calculators. By combining these approaches, students get both a conceptual and practical understanding. Also, graphical solutions help to visually interpret real-world situations modeled by quadratic equations, an essential skill in many areas of science and engineering.
Other exercises in this chapter
Problem 77
In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=\sqrt{x+1} ;
View solution Problem 77
Use the following definition. A complex number \(a+b i\) is often denoted by the letter \(z .\) Its conjugate, \(a-b i\) is denoted by \(\bar{z}\). Show that \(
View solution Problem 78
The quadratic function $$p(x)=-0.387(x-45)^{2}+2.73(x-45)-3.89$$ gives the percentage (in decimal form) of puffin eggs that hatch during a breeding season in te
View solution Problem 78
Let \(S(x)\) represent the weekly salary of a salesperson, where \(x\) is the weekly dollar amount of sales generated. If the salesperson pays \(15 \%\) of her
View solution