Problem 35 - Precalculus: Functions and Graphs

Question

Solve the equation. \(x=3+\sqrt{5 x-9}\)

Step-by-Step Solution

Answer

The solution is \(x = 9\).

Step 1: Isolate the Square Root

The given equation is \(x = 3 + \sqrt{5x - 9}\). To simplify, isolate the square root on one side if possible. In this case, it's already isolated on the right side: \(\sqrt{5x - 9} = x - 3\).

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation: \((x - 3)^2 = (\sqrt{5x - 9})^2\). This results in \((x - 3)^2 = 5x - 9\).

Step 3: Expand the Quadratic Equation

Expand the left side of the equation: \((x - 3)^2 = x^2 - 6x + 9\). So the equation becomes \(x^2 - 6x + 9 = 5x - 9\).

Step 4: Rearrange the Equation

Rearrange the equation into a standard quadratic form by setting everything to one side: \(x^2 - 6x + 9 - 5x + 9 = 0\). Simplify to get \(x^2 - 11x + 18 = 0\).

Step 5: Solve the Quadratic Equation

Use the quadratic formula to solve for \(x\), where \(a = 1\), \(b = -11\), and \(c = 18\). Substitute into \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 6: Compute the Discriminant

Calculate the discriminant: \(b^2 - 4ac = (-11)^2 - 4 \cdot 1 \cdot 18 = 121 - 72 = 49\). Since it's positive, there are two real solutions.

Step 7: Apply the Quadratic Formula

Substitute the values into the quadratic formula: \(x = \frac{-(-11) \pm \sqrt{49}}{2 \cdot 1}\). Simplify to get \(x = \frac{11 \pm 7}{2}\).

Step 8: Calculate the Solutions

Find the two potential solutions: \(x = \frac{11 + 7}{2} = 9\) and \(x = \frac{11 - 7}{2} = 2\).

Step 9: Check the Solutions

Substitute both possible solutions back into the original equation to verify: for \(x = 9\), check \(9 = 3 + \sqrt{45 - 9}\), which is true since \(9 = 3 + 6\). For \(x = 2\), check \(2 = 3 + \sqrt{10 - 9}\), which is false since \(2 eq 4\). Only \(x = 9\) is valid.

Key Concepts

Isolating the Square Root

When solving equations that involve a square root, one of the first key steps is to isolate the square root on one side of the equation. This helps simplify the process of finding the solution. In our exercise, the equation given is already in an optimal form for isolation:

• We have the equation: \(x = 3 + \sqrt{5x - 9}\).
• Rewriting, the isolated square root is \(\sqrt{5x - 9} = x - 3\).

By isolating the square root, we prepare the equation for further manipulation, like removing the square root through squaring. This step is essential since it turns the problem into a simpler form, one which can be squared on both sides to eliminate the radical.

Quadratic Formula

Once we've removed any square roots and rearranged our equation, we often end up with a quadratic equation. A quadratic equation is typically expressed in the form: \(ax^2 + bx + c = 0\). The quadratic formula is a powerful tool used to solve these equations. It is given by:

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

This formula provides solutions for \(x\) by considering the coefficients \(a\), \(b\), and \(c\) from the quadratic equation. In cases like our exercise, it aids in finding exact answers, especially when dealing with complex numbers or equations that can’t easily be factored. Here, after simplification, our quadratic equation is \(x^2 - 11x + 18 = 0\), where \(a = 1\), \(b = -11\), and \(c = 18\). Substituting these values into the formula allows us to find the solutions to the quadratic.

Understanding the Discriminant

The discriminant, found in the quadratic formula under the square root sign, is a crucial part of solving quadratic equations. The discriminant is expressed as \(b^2 - 4ac\) and gives us insight into the nature of the solutions:

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there is exactly one real solution (a repeated root).
• If it is negative, there are no real solutions, but rather two complex solutions.

In our exercise with \(b = -11\), \(a = 1\), and \(c = 18\), the discriminant is \(49\), since:

• \((b^2 - 4ac) = (-11)^2 - 4 \times 1 \times 18 = 121 - 72 = 49\).

A positive discriminant like this one indicates there are two real solutions, guiding us in the next steps of solving the quadratic.

Finding Real Solutions

After determining the nature of the solutions through the discriminant, the next step is to actually find these real solutions using the quadratic formula. For our equation, substituting \(a = 1\), \(b = -11\), and \(c = 18\) into the formula gives us:

• \(x = \frac{-(-11) \pm \sqrt{49}}{2 \times 1}\).
• This simplifies to \(x = \frac{11 \pm 7}{2}\).

This results in two potential solutions:

• \(x = \frac{18}{2} = 9\), and
• \(x = \frac{4}{2} = 2\).

However, not every mathematical solution is valid when checked back with the original equation. By substituting back, only \(x = 9\) satisfies the original equation, proving it is the actual solution. It's always crucial to verify theoretical solutions in the context of the problem.