Problem 93
Question
In this section we introduced methods of solving equations quadratic in form by substitution and solving equations involving radicals by raising each side of the equation to a power. Suppose we wish to solve $$x-\sqrt{x}-12=0$$ We can solve this equation using either of the two methods. Work Exercises \(93-96\) in order to see how both methods apply. Let \(u=\sqrt{x}\) and solve the equation by substitution. What is the value of \(u\) that does not lead to a solution of the equation?
Step-by-Step Solution
Verified Answer
The value \(u = -3\) does not lead to a solution.
1Step 1 - Substitution
Let \( u = \sqrt{x} \). Then, \( u^2 = x \). Substitute these into the original equation to get: \( u^2 - u - 12 = 0 \).
2Step 2 - Solve the Quadratic Equation
Solve the quadratic equation \( u^2 - u - 12 = 0 \) by factoring. Notice that \( -3 \cdot 4 = -12 \) and \( -3 + 4 = 1 \), so we factor the equation as \( (u - 4)(u + 3) = 0 \).
3Step 3 - Find the Roots
Set each factor equal to zero: \( u - 4 = 0 \) and \( u + 3 = 0 \). Solving these gives us \( u = 4 \) and \( u = -3 \).
4Step 4 - Verify the Solutions
Substitute back \( u = \sqrt{x} \) to check if these values lead to valid solutions for \( x \). For \( u = 4 \), we have \( x = u^2 = 16 \) and for \( u = -3 \), we have \( x = u^2 = 9 \).
5Step 5 - Check the Validity of Each Solution
Substitute \( x = 16 \) back into the original equation: \( 16 - \sqrt{16} - 12 = 0 \) simplifies to \( 16 - 4 - 12 = 0 \), which is true. Now, substitute \( x = 9 \) back into the original equation: \( 9 - \sqrt{9} - 12 = 0 \) simplifies to \( 9 - 3 - 12 eq 0 \), so it is not valid.
Key Concepts
substitution methodfactoring quadraticsradical equationsroots verification
substitution method
The substitution method is a powerful tool for solving equations, including quadratic and radical equations. It involves replacing a complex part of the equation with a simpler variable to make the equation easier to solve. In our problem, let's use the substitution method to simplify the equation:
Given equation: \( x - \sqrt{x} - 12 = 0 \)
We let \(u = \sqrt{x}\), which means \(u^2 = x\).
By substituting these into the original equation, we transform it into \( u^2 - u - 12 = 0 \).
This new equation is a quadratic equation in \(u\).
The key idea here is to pick a variable, in this case, \(u\), to simplify the parts of the original equation that are more difficult to handle directly. This helps in breaking down complex problems into simpler, more manageable steps.
Given equation: \( x - \sqrt{x} - 12 = 0 \)
We let \(u = \sqrt{x}\), which means \(u^2 = x\).
By substituting these into the original equation, we transform it into \( u^2 - u - 12 = 0 \).
This new equation is a quadratic equation in \(u\).
The key idea here is to pick a variable, in this case, \(u\), to simplify the parts of the original equation that are more difficult to handle directly. This helps in breaking down complex problems into simpler, more manageable steps.
factoring quadratics
Factoring quadratics is a method used to solve quadratic equations of the form \(ax^2 + bx + c = 0\). The idea is to express the quadratic as the product of two binomials. Let's factor the quadratic equation we obtained from the substitution method:
Given quadratic: \(u^2 - u - 12 = 0\).
We need to find two numbers that multiply to \(-12\) and add to \(-1\). These numbers are \(-3\) and \(4\).
So, we factor the quadratic as follows:
\((u - 4)(u + 3) = 0\)
By factoring, we transformed the quadratic equation into a product of two simpler linear equations. Each of these factors can be set to zero to find the possible values of \(u\).
Given quadratic: \(u^2 - u - 12 = 0\).
We need to find two numbers that multiply to \(-12\) and add to \(-1\). These numbers are \(-3\) and \(4\).
So, we factor the quadratic as follows:
\((u - 4)(u + 3) = 0\)
By factoring, we transformed the quadratic equation into a product of two simpler linear equations. Each of these factors can be set to zero to find the possible values of \(u\).
radical equations
A radical equation is an equation in which the variable appears under a radical sign (such as the square root). Solving such equations often involves isolating the radical and then squaring both sides to eliminate it.
In our problem, the original equation was a radical equation: \( x - \sqrt{x} - 12 = 0 \).
By making the substitution \(u = \sqrt{x}\), we simplified the radical equation into a quadratic one.
However, it's important to remember that when we square both sides, we might introduce extraneous solutions that don't satisfy the original equation. So, we must always verify our solutions.
After finding \(u = 4\) and \(u = -3\), we need to substitute back into \(u = \sqrt{x}\) to find \(x\) and check which solutions are valid.
In our problem, the original equation was a radical equation: \( x - \sqrt{x} - 12 = 0 \).
By making the substitution \(u = \sqrt{x}\), we simplified the radical equation into a quadratic one.
However, it's important to remember that when we square both sides, we might introduce extraneous solutions that don't satisfy the original equation. So, we must always verify our solutions.
After finding \(u = 4\) and \(u = -3\), we need to substitute back into \(u = \sqrt{x}\) to find \(x\) and check which solutions are valid.
roots verification
Roots verification is a crucial step in solving equations, especially when using substitution and factoring, as we might get extraneous solutions. To verify the roots, we must substitute them back into the original equation and check if they satisfy it.
In this problem, we found \(u = 4\) and \(u = -3\):
For \(u = 4\):
Substituting into \(u = \sqrt{x}\) gives \(x = 16\).
Check: \(16 - \sqrt{16} - 12 = 0\) simplifies to \(16 - 4 - 12 = 0\), which is true.
For \(u = -3\):
Substituting into \(u = \sqrt{x}\) gives \(x = 9\).
Check: \(9 - \sqrt{9} - 12 = 0\) simplifies to \(9 - 3 - 12 = -6\), which is not zero and thus, not valid.
By verifying the roots, we ensure that only true solutions to the original equation are considered.
In this problem, we found \(u = 4\) and \(u = -3\):
For \(u = 4\):
Substituting into \(u = \sqrt{x}\) gives \(x = 16\).
Check: \(16 - \sqrt{16} - 12 = 0\) simplifies to \(16 - 4 - 12 = 0\), which is true.
For \(u = -3\):
Substituting into \(u = \sqrt{x}\) gives \(x = 9\).
Check: \(9 - \sqrt{9} - 12 = 0\) simplifies to \(9 - 3 - 12 = -6\), which is not zero and thus, not valid.
By verifying the roots, we ensure that only true solutions to the original equation are considered.
Other exercises in this chapter
Problem 93
Simplify each power of i. $$i^{-13}$$
View solution Problem 93
Solve each problem. The industrial process that is used to convert methanol to gasoline is carried out at a temperature range of \(680^{\circ} \mathrm{F}\) to \
View solution Problem 93
Use a graphing calculator to graph each function defined as follows, using the given viewing window. Use the graph to decide which functions are one-to-one. If
View solution Problem 94
Simplify each power of i. $$i^{-14}$$
View solution