Problem 105 - College Algebra

Question

Which one of the following is true? a. Squaring both sides of \(\sqrt{y+4}+\sqrt{y-1}=5\) leads to \(y+4+y-1=25,\) an equation with no radicals. b. The equation \(\left(x^{2}-2 x\right)^{9}-5\left(x^{2}-2 x\right)^{3}+6=0\) is quadratic in form and should be solved by letting \(t=\left(x^{2}-2 x\right)^{3}\) c. If a radical equation has two proposed solutions and one of these values is not a solution, the other value is also not a solution. d. None of these statements is true.

Step-by-Step Solution

Answer

b. The equation \((x^{2}-2 x)^{9}-5(x^{2}-2 x)^{3}+6=0\) is quadratic in form and should be solved by letting \(t=\left(x^{2}-2 x\right)^{3}\) is the correct statement.

Step 1: Evaluate Statement a

Statement a suggests that squaring both sides of the equation \(\sqrt{y+4}+\sqrt{y-1} = 5\) leads to a simplified equation \(y+4+y-1 = 25\). Upon squaring, the equation would result in something a bit more complex since the addition inside the square root signs will be distributed leading to a sum of roots rather than this simplified equation. Therefore, statement a is false.

Step 2: Evaluate Statement b

Statement b indicates that the equation \((x^{2}-2x)^{9}-5(x^{2}-2x)^{3}+6=0\) is quadratic in form by letting \(t = (x^{2}-2x)^{3}\). By making this substitution, it simplifies the equation into a quadratic form \(t^{3}-5t+6=0\). Therefore, statement b is true.

Step 3: Evaluate Statement c

Statement c suggests that in a radical equation, if one of two proposed solutions is not a solution, then the other one will also not be a solution. This is not inherently true. Radical equations can have one or more solutions and having one solution doesn't impact the validity of any other potential solutions. Therefore, statement c is also false.

Key Concepts

Quadratic Form

In mathematics, a quadratic form refers to any expression where variables are squared, but no higher powers are included. This concept allows us to transform complex equations into simpler, solvable forms. If you see terms like \(x^2\) or \(ax^2 + bx + c\), you're likely dealing with a quadratic form.
In the exercise provided, the expression \((x^2 - 2x)^9 - 5(x^2 - 2x)^3 + 6 = 0\) appears complex. However, by spotting a structured form, we simplify it using substitution. Here, we identified \((x^2 - 2x)^3\) as \(t\).
Once this substitution is made, the equation becomes \(t^3 - 5t + 6 = 0\), which is a quadratic form. This step reduces complexity and makes solving for \(x\) much easier. Understanding quadratic forms can streamline complex algebra problems, turning them into manageable equations.

Substitution Method

The substitution method is a powerful tool for solving equations, especially when they seem complex at first glance. In this method, we replace part of the equation with a simpler variable, making the problem easier to handle.

The exercise demonstrated this with the equation \((x^2 - 2x)^9 - 5(x^2 - 2x)^3 + 6 = 0\). By introducing \(t = (x^2 - 2x)^3\), we simplified the original equation to \(t^3 - 5t + 6 = 0\). Now, instead of dealing directly with a tricky expression like \((x^2 - 2x)\), we work with a simpler quadratic equation in \(t\).

• This approach allows us to focus on solving the quadratic equation \(t^3 - 5t + 6 = 0\) without distractions.
• Once solved, we backtrack to find the original variable \(x\) by solving \((x^2 - 2x)^3 = t\).

This method turns step-by-step complexity into a more straightforward task.

Solution Verification

Even after successfully finding potential solutions to an equation, verifying these solutions is crucial. Without verification, there might be errors or extraneous solutions – solutions that emerge through the solving process but do not actually satisfy the original equation.
In the exercise, statement c challenges us to consider the importance of verifying each proposed solution in a radical equation. Although one solution might not work, this does not preclude other solutions from being valid.

• Always substitute each solution back into the original equation.
• Check for any inconsistencies or errors in the results.

Solution verification ensures accuracy and confirms the final answers are correct within the context of the problem.

Simplifying Equations

Simplifying equations is a vital step in solving mathematical problems. It involves reducing the equation to its most basic form, which can often reveal the solution pathway.
Statement a in the exercise stresses the importance of proper simplification. It incorrectly asserts that squaring \(\sqrt{y+4} + \sqrt{y-1} = 5\) leads directly to \(y + 4 + y - 1 = 25\).

However, when squaring an equation with radicals, one should consider using the distributive property. Correctly, upon squaring both sides:

• You should expand as \((\sqrt{y+4} + \sqrt{y-1})^2 = 25\).
• This leads to \( (\sqrt{y+4})^2 + 2\sqrt{(y+4)(y-1)} + (\sqrt{y-1})^2 = 25\), which simplifies further with careful calculation.

Proper simplification is not just about removing radicals but about restructuring the equation fully and accurately.