Problem 86 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$8(x-4)^{4}-10(x-4)^{2}=-3$$

Step-by-Step Solution

Answer

The solutions for x are \( x = 4 \pm \frac{\sqrt{3}}{2}\) and \( x = 4 \pm \frac{\sqrt{2}}{2}\).

Step 1 - Let substitution

Letting substitution help simplify the equation. Set \(u = (x - 4)^2\). The equation then becomes an equation in terms of \(u\): \[8u^2 - 10u = -3\].

Step 2 - Arrange the equation

Rearrange the equation to standard quadratic form: \[8u^2 - 10u + 3 = 0\].

Step 3 - Solve the quadratic equation

Solve the quadratic equation using the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 8\), \(b = -10\), and \(c = 3\): \[ u = \frac{10 \pm \sqrt{(-10)^2 - 4(8)(3)}}{2(8)} = \frac{10 \pm \sqrt{100 - 96}}{16} = \frac{10 \pm 2}{16} \].

Step 4 - Find the roots of u

Evaluate the roots: \(u = \frac{10 + 2}{16} = \frac{12}{16} = \frac{3}{4}\) and \(u = \frac{10 - 2}{16} = \frac{8}{16} = \frac{1}{2}\). Thus, \(u = \frac{3}{4}\) and \(u = \frac{1}{2}\).

Step 5 - Reverse the substitution

Recall that \(u = (x - 4)^2\). Set \( (x - 4)^2 = \frac{3}{4} \) and \( (x - 4)^2 = \frac{1}{2} \).

Step 6 - Solve for x

Solve these equations for \(x\): \((x - 4)^2 = \frac{3}{4}\)\[(x - 4) = \pm \sqrt{\frac{3}{4}}\]\( x - 4 = \pm \frac{\sqrt{3}}{2} \)\( x = 4 \pm \frac{\sqrt{3}}{2}\). \((x - 4)^2 = \frac{1}{2}\)\[(x - 4) = \pm \sqrt{\frac{1}{2}}\]\( x - 4 = \pm \frac{1}{\sqrt{2}} \)\( x = 4 \pm \frac{\sqrt{2}}{2}\).

Key Concepts

substitution method

A useful technique for solving complex equations is the substitution method.
This method simplifies an otherwise difficult problem by introducing a new variable.
In this exercise, we start by letting \(u = (x-4)^2\).
This substitution transforms our original equation into something more manageable: \[8u^2 - 10u = -3\].
Now our goal is to solve for \(u\), which is a simpler task.
After finding \(u\), we reverse the substitution to find the original variable \(x\).
This method helps break down the problem into smaller and more straightforward steps.
Whether the equation is polynomial or otherwise complicated, the substitution method can often simplify the solving process significantly.

quadratic formula

The quadratic formula is a powerful tool for solving quadratic equations.
If you have an equation of the form \(ax^2 + bx + c = 0\), you can find the solutions using the formula:
\[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this particular exercise, we applied the quadratic formula to solve for \(u\) in the equation \(8u^2 - 10u + 3 = 0\).
Here, we identified our coefficients as \(a = 8\), \(b = -10\), and \(c = 3\).
This gave us the equation:
\[ u = \frac{10 \pm \sqrt{100 - 96}}{16} \]
which simplifies to \( u = \frac{10 \pm 2}{16}\).
Thus, we found our roots to be \( u = \frac{3}{4} \) and \( u = \frac{1}{2} \).
To solve quadratic equations systematically, make sure to correctly identify \(a\), \(b\), and \(c\) from first principles, and then plug them into the quadratic formula.

simplification

Simplification is a crucial part of solving any mathematical problem.
It helps to make the equation more manageable and easier to solve.
In this exercise, we started with the equation \(8(x-4)^4 - 10(x-4)^2 = -3\).
We used the substitution \(u = (x-4)^2\) to simplify it to a quadratic form: \(8u^2 - 10u + 3 = 0\).
After solving for \(u\) with the quadratic formula, we got two possible values: \(u = \frac{3}{4}\) and \(u = \frac{1}{2}\).
Next, we reversed the substitution to find \(x\) by solving: \((x-4)^2 = \frac{3}{4}\) and \((x-4)^2 = \frac{1}{2}\).
Simplifying further, we obtained: \[x = 4 \pm \frac{\sqrt{3}}{2}\] and \[x = 4 \pm \frac{\sqrt{2}}{2}\].
Thus, simplification made it possible to break down a complex fourth-degree equation into simpler, manageable quadratic components.