Problem 44 - Algebra A Combined Function

Question

Solve. $$ \sqrt{2 x-1}-4=-\sqrt{x-4} $$

Step-by-Step Solution

Answer

No solution, as neither potential solution satisfies the original equation.

Step 1: Isolate one of the square roots

Firstly, let's move the '-4' to the other side of the equation to isolate one of the square roots. The equation becomes: \[\sqrt{2x-1} = 4 - \sqrt{x-4}\].

Step 2: Square both sides of the equation

To eliminate the square roots, square both sides of the equation. This results in: \[(\sqrt{2x-1})^2 = (4 - \sqrt{x-4})^2\] which simplifies to: \[2x-1 = 16 - 8\sqrt{x-4} + (x-4)\].

Step 3: Simplify and rearrange the equation

Simplify the equation from Step 2. It becomes: \[2x - 1 = x + 12 - 8\sqrt{x-4}\] Now, combine like terms to get: \[x - 13 = -8\sqrt{x-4}\].

Step 4: Isolate the remaining square root

Move the terms to isolate the square root. The equation becomes: \[x - 13 = -8\sqrt{x-4}\] then divide both sides by -8 to get: \[\sqrt{x-4} = \frac{13-x}{8}\].

Step 5: Square both sides again

Square both sides again to eliminate the square root: \[(\sqrt{x-4})^2 = \left(\frac{13-x}{8}\right)^2\] which simplifies to: \[x-4 = \frac{(13-x)^2}{64}\].

Step 6: Multiply through and expand terms

Multiply both sides by 64 to clear the fraction: \[64(x - 4) = (13-x)^2\] Expand the right side: \[64x - 256 = 169 - 26x + x^2\].

Step 7: Set equation to zero

Rearrange the equation to gather terms on one side, resulting in: \[x^2 - 90x + 425 = 0\].

Step 8: Solve quadratic equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve: where \(a=1\), \(b=-90\), \(c=425\). Calculate the discriminant: \(b^2 - 4ac = 8100 - 1700 = 6400\). Then \(x = \frac{90 \pm 80}{2}\), giving solutions: \(x = 85\) and \(x = 5\).

Step 9: Check solutions in the original equation

Substitute \(x = 85\) and \(x = 5\) back into the original equation to verify. For \(x = 85\), \(\sqrt{169} - 4 = -\sqrt{81}\) which is false. For \(x = 5\), \(\sqrt{9} - 4 = -\sqrt{1}\) which is also false.

Key Concepts

Square Roots

Square roots are one of the essential concepts in algebra, appearing frequently in various equations. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because

• \(4 \times 4 = 16\).

Square roots are often represented by the radical symbol \(\sqrt{}\). These can sometimes involve numbers under the radical being simplified, especially when dealing with algebraic expressions. Understanding how to manipulate and simplify square roots is critical for solving equations that contain them. In the exercise, square roots are overcome by squaring both sides of the equation, which helps remove the radicals and simplifies solving.

Quadratic Equations

Quadratic equations are polynomial equations of degree 2, generally taking the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. These equations can have one, two, or no real solutions depending on the discriminant's value, which is calculated as \(b^2 - 4ac\). When solving quadratic equations, it's crucial to bring all terms to one side, setting the equation to zero. This step is highlighted in the exercise where we rearrange the terms to form a quadratic equation in Step 7. The resulting equation \(x^2 - 90x + 425 = 0\) illustrates how quadratic expressions can emerge from more complex problems.

Quadratic Formula

The quadratic formula is a powerful tool that provides a straightforward method to find the solutions of a quadratic equation. This formula is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation. It's particularly useful when factoring is not easy or possible. Its derivation comes from completing the square on the general quadratic equation. In the exercise, we use this formula in Step 8 to solve \(x^2 - 90x + 425 = 0\) by calculating the discriminant first. The result, given as \(b^2 - 4ac\), determines the solutions - real and distinct if the discriminant is positive.

Equation Solving

Solving equations is a fundamental skill in algebra that involves finding the value(s) of the variable that satisfy the equation. The problem presented in the exercise involves isolating terms, dealing with square roots, and manipulating algebraic expressions. Here's a step-by-step approach:

• First, isolate one of the terms, such as by moving constant terms to the other side.
• Utilize techniques such as squaring to simplify expressions and eliminate square roots, as seen in the initial steps of the solution.
• Reorganize the equation, combining like terms to simplify further.
• If a quadratic equation arises, solve it using methods like the quadratic formula.

After finding solutions, always substitute them back into the original equation to verify their correctness. This helps ensure that any extraneous solutions introduced during squaring are identified.