Solving Miscellaneous Equations: Square Roots, Rational Exponents, Quadratic Type, and Absolute Values

Section 1.6: Miscellaneous Equations

Equations Involving Square Roots

Equations containing square roots require special techniques for solving. The main method is to isolate the square root and then square both sides to eliminate it. However, squaring both sides can introduce extraneous solutions, so it is essential to check all solutions in the original equation.

• Key Point 1: Isolate the square root on one side before squaring both sides.

• Key Point 2: Squaring both sides may introduce extraneous solutions; always check your answers in the original equation.

• Key Point 3: If there are two square roots, isolate one on each side before squaring.

• Example: Solve $\sqrt{x} = 3$.

  • Square both sides: $\left(\sqrt{x}\right)^2 = 3^2$

  • $x = 9$

  • Check: $\sqrt{9} = 3$ (valid solution)

• Example: Solve $\sqrt{x} = -3$.

  • Square both sides: $\left(\sqrt{x}\right)^2 = (-3)^2$

  • $x = 9$

  • Check: $\sqrt{9} = 3 \neq -3$ (extraneous solution)

• Example: Solve $\sqrt{x} + 2 = x$.

  • Isolate: $\sqrt{x} = x - 2$

  • Square both sides: $x = (x - 2)^2$

  • Expand: $x = x^2 - 4x + 4$

  • Rearrange: $x^2 - 5x + 4 = 0$

  • Solve quadratic: $x = 4$ or $x = 1$

  • Check both in original equation.

Additional info: Extraneous solutions are common when raising both sides to an even power. Always check solutions in the original equation.

Equations with Rational Exponents

Equations with rational exponents can be solved by raising both sides to the reciprocal of the exponent. The expression $x^{m/n}$ means the nth root of x raised to the mth power. When taking even roots, consider both positive and negative solutions.

• Key Point 1: $x^{m/n} = k$ is equivalent to $\left(\sqrt[n]{x}\right)^m = k$.

• Key Point 2: Raise both sides to the reciprocal of the exponent to solve for x.

• Key Point 3: If the root is even, include both $+$ and $-$ solutions; if odd, only the principal root.

• Example: Solve $x^{2/3} = 4$.

  • Raise both sides to $3/2$: $\left(x^{2/3}\right)^{3/2} = 4^{3/2}$

  • $x = \pm \left(\sqrt{4}\right)^3 = \pm 8$

• Example: Solve $x^{4/3} = 625$.

  • Raise both sides to $3/4$: $x = \pm \left(\sqrt[4]{625}\right)^3$

  • $\sqrt[4]{625} = 5$; $x = \pm 125$

• Example: Solve $x^{-1/2} = \sqrt{2}$.

  • Raise both sides to $-2$: $x = \left(\sqrt{2}\right)^{-2} = \frac{1}{2}$

Additional info: Always check for extraneous solutions, especially when raising both sides to an exponent.

Equations of Quadratic Type

Some equations can be rewritten as quadratic equations by substituting a new variable for a more complex expression. After solving the quadratic, substitute back and solve for the original variable.

• Key Point 1: Substitute $u$ for a complicated expression to convert to quadratic form.

• Key Point 2: Solve the quadratic equation for $u$, then substitute back and solve for the original variable.

• Example: Solve $x^4 + 14x^2 - 32 = 0$.

  • Let $u = x^2$, so $u^2 + 14u - 32 = 0$

  • Solve quadratic: $u = 2$ or $u = -16$

  • Substitute back: $x^2 = 2 \Rightarrow x = \pm \sqrt{2}$; $x^2 = -16$ (no real solution)

• Example: Solve $x^{2/3} - 9x^{1/3} + 8 = 0$.

  • Let $u = x^{1/3}$, so $u^2 - 9u + 8 = 0$

  • Solve quadratic: $u = 8$ or $u = 1$

  • Substitute back: $x^{1/3} = 8 \Rightarrow x = 512$; $x^{1/3} = 1 \Rightarrow x = 1$

Additional info: This method is useful for equations involving powers or roots that can be grouped into quadratic form.

Equations Involving Absolute Values

Absolute value equations are solved by isolating the absolute value and then considering both the positive and negative cases. The definition of absolute value is:

• $|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

• Key Point 1: Isolate the absolute value on one side.

• Key Point 2: Remove the absolute value by considering both cases: $x$ and $-x$.

• Key Point 3: If variables appear outside the absolute value, check all solutions in the original equation.

• Example: Solve $2|x + 8| - 6 = 0$.

  • Isolate: $2|x + 8| = 6 \Rightarrow |x + 8| = 3$

  • Cases: $x + 8 = 3 \Rightarrow x = -5$; $x + 8 = -3 \Rightarrow x = -11$

• Example: Solve $|3x + 4| = 12$.

  • Cases: $3x + 4 = 12 \Rightarrow x = \frac{8}{3}$; $3x + 4 = -12 \Rightarrow x = -\frac{16}{3}$

• Example: Solve $|x^2 - 6| = 5x$.

  • Cases: $x^2 - 6 = 5x \Rightarrow x^2 - 5x - 6 = 0$

  • $x^2 - 6 = -5x \Rightarrow x^2 + 5x - 6 = 0$

  • Solve both quadratics and check solutions in the original equation.

• Key Point 4: When two absolute values are equal, set the expressions inside equal and opposite.

  • Example: $|x| = |x - 1|$

  • Cases: $x = x - 1$ (no solution); $x = -(x - 1)$

  • Solve: $x = -x + 1 \Rightarrow 2x = 1 \Rightarrow x = \frac{1}{2}$

Additional info: Always check solutions, especially when variables are outside the absolute value or when two absolute values are set equal.

Summary Table: Methods for Solving Miscellaneous Equations

Equation Type	Main Method	Check for Extraneous Solutions?	Example
Square Roots	Isolate root, square both sides	Yes	$\sqrt{x} = 3$
Rational Exponents	Raise both sides to reciprocal exponent	Yes	$x^{2/3} = 4$
Quadratic Type	Substitute variable, solve quadratic	Yes	$x^4 + 14x^2 - 32 = 0$
Absolute Value	Split into two cases, solve each	Yes	$|x + 8| = 3$

Additional info: The table summarizes the main methods and the importance of checking for extraneous solutions in each type.