Problem 44 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \( x = 5 \).

Step 1: Isolate the Square Roots

The equation is \( \sqrt{2x - 1} - 4 = -\sqrt{x-4} \). First, let's move one of the square roots to the other side by adding \( \sqrt{x-4} \) to both sides. This gives us: \( \sqrt{2x - 1} = \sqrt{x - 4} + 4 \).

Step 2: Square Both Sides of the Equation

To eliminate the square roots, square both sides of the equation: \( (\sqrt{2x - 1})^2 = (\sqrt{x - 4} + 4)^2 \). Simplifying gives: \( 2x - 1 = (x - 4) + 8 \sqrt{x-4} + 16 \).

Step 3: Simplify the Equation

Combine like terms: \( 2x - 1 = x + 12 + 8 \sqrt{x-4} \). Move \( x \) to the left side to get: \( x - 1 = 12 + 8 \sqrt{x-4} \). Then, isolate the square root term: \( x - 13 = 8 \sqrt{x-4} \).

Step 4: Divide and Square Again

Divide both sides by 8: \( \frac{x - 13}{8} = \sqrt{x - 4} \). Square both sides again to eliminate the square root: \( \left(\frac{x - 13}{8}\right)^2 = x - 4 \). Simplify by expanding and rearranging: \( \frac{(x - 13)^2}{64} = x - 4 \).

Step 5: Cross Multiply and Expand

Multiply through by 64 to eliminate the fraction: \( (x - 13)^2 = 64(x - 4) \). Expand both sides: \( x^2 - 26x + 169 = 64x - 256 \).

Step 6: Rearrange and Solve the Quadratic Equation

Move all terms to one side to form a quadratic equation: \( x^2 - 90x + 425 = 0 \). Solve the quadratic using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -90, c = 425 \).

Step 7: Calculate Discriminant and Roots

Calculate the discriminant: \( b^2 - 4ac = 8100 - 1700 = 6400 \). The roots are \( x = \frac{90 \pm 80}{2} \). Thus, \( x = 85 \) or \( x = 5 \).

Step 8: Verify Roots in Original Equation

Check each solution in the original equation to ensure they work. For \( x = 85 \), \( \sqrt{2(85) - 1} - 4 = -\sqrt{85 - 4} \) simplifies to \( 13 - 4 = -9 \). For \( x = 5 \), \( \sqrt{2(5) - 1} - 4 = -\sqrt{5 - 4} \) simplifies to \( -1 \). Only \( x = 5 \) satisfies both sides of the original equation.

Key Concepts

Square Roots

Square roots are mathematical expressions that represent a value which, when multiplied by itself, gives the original number. They are symbolized by the radical sign \( \sqrt{} \).
For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \). When working with equations that include square roots, one common goal is to remove the square root sign by squaring both sides of the equation.

• This often simplifies the equation and leaves a polynomial expression.
• Be cautious, though, because squaring both sides can introduce extraneous solutions.

In the context of solving radical equations like \( \sqrt{2x - 1} = \sqrt{x - 4} + 4 \), the process involves ensuring that each square root term is isolated before squaring each side to help eliminate the roots. By isolating one of the radicals, often with simple algebraic manipulation, you can then square both sides clearly and cohesively.
This process helps in reaching a more manageable form, where conventional solving methods for quadratic equations can be applied.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable \( x \) with the general form \( ax^2 + bx + c = 0 \). These equations appear frequently in various algebra problems and have a characteristic curve called a parabola.

• The standard approach to solve quadratic equations is using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \),
• Although sometimes factoring or completing the square might be applicable depending on its complexity.

To form a quadratic equation from a scenario involving radicals like in our problem, squaring the radicals typically presents terms up to the second degree, which then require arranging all terms as a unified quadratic equation. In the exercise, squaring and simplifying expressions resulted in an equation \( x^2 - 90x + 425 = 0 \). Solving these requires calculated application of the quadratic formula or, occasionally, recognizing patterns that allow easier solutions.

Discriminant

The discriminant is a key element in understanding the nature of solutions for a quadratic equation. Calculated as \( \Delta = b^2 - 4ac \), it tells us about the number and nature of the roots of the equation:

• If \( \Delta > 0 \), the quadratic equation has two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root (it's a perfect square trinomial).
• If \( \Delta < 0 \), the equation has no real roots, but two complex roots.

In our exercise with \( x^2 - 90x + 425 = 0 \), the discriminant was calculated to be 6400, which is greater than zero. This indicates two real roots exist for this equation.
Using these roots and re-substituting them into the original equation can help verify which, if any, fit the context of the problem. Only roots that satisfy the original given radical equation are valid; hence verification becomes crucial.