Problem 88
Question
Solve each equation. $$7 x^{-2}-10 x^{-1}-8=0$$
Step-by-Step Solution
Verified Answer
x = \frac{1}{2} or x = -\frac{7}{4}
1Step 1: Substitute Variables
To simplify the equation, let's substitute a new variable. Let: \[ y = x^{-1} \] This changes the equation from: \[ 7 x^{-2} - 10 x^{-1} - 8 = 0 \] to \[ 7y^2 - 10y - 8 = 0 \]
2Step 2: Solve the Quadratic Equation
Now, solve the quadratic equation \( 7y^2 - 10y - 8 = 0 \) using the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 7 \), \( b = -10 \), and \( c = -8 \). Substituting the values, we get: \[ y = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 7 \cdot (-8)}}{2 \cdot 7} \] Simplifying further: \[ y = \frac{10 \pm \sqrt{100 + 224}}{14} \] \[ y = \frac{10 \pm \sqrt{324}}{14} \] \[ y = \frac{10 \pm 18}{14} \]
3Step 3: Calculate the Roots
Calculate the two possible values for \( y \): \[ y_1 = \frac{10 + 18}{14} = \frac{28}{14} = 2 \] \[ y_2 = \frac{10 - 18}{14} = \frac{-8}{14} = - \frac{4}{7} \]
4Step 4: Substitute Back to Solve for x
Recall that \( y = x^{-1} \), thus: \[ x^{-1} = 2 \Rightarrow x = \frac{1}{2} \] and \[ x^{-1} = -\frac{4}{7} \Rightarrow x = - \frac{7}{4} \]
Key Concepts
quadratic equationsubstitution methodsolving for x
quadratic equation
In the given exercise, the equation initially involves a term with an inverse square of a variable. By making a substitution, this complex equation is converted into a quadratic equation.
A quadratic equation is any equation that can be written in the form: \[ ax^2 + bx + c = 0 \]
where \( a \), \( b \), and \( c \) are constants, and the variable is squared. In our case, after the substitution \( y = x^{-1} \), we have: \( 7y^2 -10y -8 = 0 \).
Quadratic equations have a characteristic 'U' shaped graph called a parabola. To solve them, we often use the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula provides solutions by considering the discriminant \( b^2 - 4ac \), which helps determine the nature of the roots (real or complex).
Understanding the quadratic equation and its solutions is key to solving the given exercise efficiently.
A quadratic equation is any equation that can be written in the form: \[ ax^2 + bx + c = 0 \]
where \( a \), \( b \), and \( c \) are constants, and the variable is squared. In our case, after the substitution \( y = x^{-1} \), we have: \( 7y^2 -10y -8 = 0 \).
Quadratic equations have a characteristic 'U' shaped graph called a parabola. To solve them, we often use the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula provides solutions by considering the discriminant \( b^2 - 4ac \), which helps determine the nature of the roots (real or complex).
Understanding the quadratic equation and its solutions is key to solving the given exercise efficiently.
substitution method
The substitution method simplifies complex equations by introducing a new variable. In our exercise, we use it to reduce a complex rational equation into a more manageable quadratic one.
First, let's substitute \( y = x^{-1} \).
Why do we do this? Because dealing with \( x^{-2} \) and \( x^{-1} \) directly can be challenging.
Using substitution: \[ 7x^{-2} - 10x^{-1} - 8 = 0 \]
transforms into: \[ 7y^2 -10 y -8 = 0 \].
Now, the new variable simplifies our problem to a recognizable form — a quadratic equation.
Once we solve for \( y \), we revert to our original variable by replacing \( y \) back with \( x^{-1} \).
This method is powerful for breaking down and solving challenging equations.
First, let's substitute \( y = x^{-1} \).
Why do we do this? Because dealing with \( x^{-2} \) and \( x^{-1} \) directly can be challenging.
Using substitution: \[ 7x^{-2} - 10x^{-1} - 8 = 0 \]
transforms into: \[ 7y^2 -10 y -8 = 0 \].
Now, the new variable simplifies our problem to a recognizable form — a quadratic equation.
Once we solve for \( y \), we revert to our original variable by replacing \( y \) back with \( x^{-1} \).
This method is powerful for breaking down and solving challenging equations.
solving for x
After converting our complex equation to a simpler form and solving it, we need to revert to our original variable.
In the given exercise, after solving the quadratic equation \( 7y^2 - 10y - 8 = 0 \), we find the roots: \[ y_1 = 2 \] and \[ y_2 = -\frac{4}{7} \].
Recall the substitution \( y = x^{-1} \), which means: \[ x^{-1} = y \].
Substituting the values of \( y \) back: \[ x^{-1} = 2 \Rightarrow x = \frac{1}{2} \]
and \[ x^{-1} = -\frac{4}{7} \Rightarrow x = - \frac{7}{4} \].
This reversion step is crucial as it translates our solutions back to the context of the original problem.
Always make sure to interpret the final roots correctly in terms of the original variable.
In the given exercise, after solving the quadratic equation \( 7y^2 - 10y - 8 = 0 \), we find the roots: \[ y_1 = 2 \] and \[ y_2 = -\frac{4}{7} \].
Recall the substitution \( y = x^{-1} \), which means: \[ x^{-1} = y \].
Substituting the values of \( y \) back: \[ x^{-1} = 2 \Rightarrow x = \frac{1}{2} \]
and \[ x^{-1} = -\frac{4}{7} \Rightarrow x = - \frac{7}{4} \].
This reversion step is crucial as it translates our solutions back to the context of the original problem.
Always make sure to interpret the final roots correctly in terms of the original variable.
Other exercises in this chapter
Problem 88
Write each statement as an absolute value equation or inequality. \(q\) is no more than 8 units from 22.
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Solve each rational inequality. Write each solution set in interval notation.4 $$\frac{9 x-8}{4 x^{2}+25}
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Evaluate the discriminant for each equation. Then use it to predict the number of distinct solutions, and whether they are rational, irrational, or non real com
View solution Problem 89
Simplify each power of i. $$i^{23}$$
View solution