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Magazine Chapter 10: Problem 83
Without graphing, determine the x-intercepts of the graphs given by each of the following. $$g(x)=6 x^{1 / 2}+6 x^{-1 / 2}-37$$
Short Answer
Expert verified
The x-intercepts are at \( x = 36 \) and \( x = \frac{1}{36} \).
Step by step solution
01
Set the function equal to zero
To find the x-intercepts, set the function equal to zero, because the x-intercepts occur where the graph crosses the x-axis, i.e., where the function's value is zero.\[ g(x) = 6x^{1/2} + 6x^{-1/2} - 37 = 0 \]
02
Substitute variables to simplify
Let鈥檚 make a substitution to simplify the equation. Set \( y = x^{1/2} \). This means that \( x^{-1/2} = \frac{1}{y} \), so the equation becomes:\[ 6y + 6\frac{1}{y} - 37 = 0 \]
03
Clear the fraction
Multiply both sides of the equation by \( y \) to clear the fraction:\[ 6y^2 + 6 - 37y = 0 \]This simplifies to:\[ 6y^2 - 37y + 6 = 0 \]
04
Solve the quadratic equation
Solve the quadratic equation \( 6y^2 - 37y + 6 = 0 \) using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):Where \( a = 6 \), \( b = -37 \), and \( c = 6 \), so\[ y = \frac{37 \pm \sqrt{(-37)^2 - 4(6)(6)}}{2(6)} \]Simplify under the square root:\[ y = \frac{37 \pm \sqrt{1369 - 144}}{12} \]\[ y = \frac{37 \pm \sqrt{1225}}{12} \]\[ y = \frac{37 \pm 35}{12} \]
05
Find the values of y
Calculate the two values for \( y \):\[ y = \frac{37 + 35}{12} = \frac{72}{12} = 6 \]\[ y = \frac{37 - 35}{12} = \frac{2}{12} = \frac{1}{6} \]
06
Back-substitute to find x
Recall that \( y = x^{1/2} \). Back-substitute to find the values of \( x \):\[ 1. \ y = 6, \ x = y^2 = 6^2 = 36 \]\[ 2. \ y = \frac{1}{6}, \ x = y^2 = \left(\frac{1}{6}\right)^2 = \frac{1}{36} \]
07
Conclusion
The x-intercepts of the function are at \( x = 36 \) and \( x = \frac{1}{36} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equation
A quadratic equation is one of the fundamental concepts in algebra. It comes in the standard form: \[ ax^2 + bx + c = 0 \] Here, *a*, *b*, and *c* are constants, and *x* represents the variable. The graph of a quadratic equation forms a parabola, which may open upwards (if *a* is positive) or downwards (if *a* is negative). Quadratic equations can be solved using several methods including factoring, completing the square, and the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. In the given problem, we transform the simplified form into a quadratic equation, making it easier to solve using the quadratic formula. This demonstrates a vital technique for students to master in order to efficiently tackle complex algebraic problems.
Function Simplification
Simplifying a function means expressing it in a reduced form that is easier to work with. This often involves combining like terms, factoring, or making substitutions. In the given exercise, we simplify the original function \[ g(x) = 6x^{1/2} + 6x^{-1/2} - 37 = 0 \] by substituting *y* for *x*^1/2, translating the equation into a more familiar quadratic form. Simplification reduces complexity and transforms the problem into a standard form, which is easier to solve.
Substitution Method
The substitution method involves replacing a variable in an equation with another expression to make it easier to solve. In our exercise, we set \[ y = x^{1/2} \]. This converts the initial equation into \[ 6y + 6\frac{1}{y} - 37 = 0 \]. Then, by multiplying through by *y* to clear the fraction, we get a quadratic equation. Substitution is a valuable tool in algebra as it allows complicated expressions to be simplified, facilitating easier and more efficient problem-solving.
Algebraic Problem-Solving
Solving algebraic problems involves various techniques and methods to find unknown values. Here's a structured approach to solving the problem at hand:
- Set the function equal to zero to find the x-intercepts.
- Use substitution to transform a complex equation into a simpler form.
- Clear fractions by multiplying through by the necessary term.
- Solve the resulting quadratic equation using the quadratic formula.
- Back-substitute the obtained values to solve for the original variable.
- Finalize the solution by identifying the x-intercepts.
