Problem 84 - Beginning and Intermediate Algebra

Question

Solve each equation. (All solutions are nonreal complex numbers.) $$ (4 m-7)^{2}=-27 $$

Step-by-Step Solution

Answer

(3i√3 + 7)/4 and (-3i√3 + 7)/4

Step 1 - Isolate the squared term

The given equation is (4m - 7)^2 = -27.

Step 2 - Take the square root of both sides

To solve for the term inside the square, take the square root of both sides. (4m - 7) = ± √(-27).

Step 3 - Simplify the square root

Simplify ± √(-27) as ±3 i√3.

Step 4 - Solve for the variable

Set up two separate equations: (4m - 7) = 3 i√3 and (4m - 7) = -3 i√3.

Step 5 - Solve each equation

For the first equation: 4m = 3 i√3 + 7 m = (3 i√3 + 7)/4. For the second equation: 4m = -3 i√3 + 7 m = (-3 i√3 + 7)/4. Both solutions can be written as:

Key Concepts

isolating the squared term

To begin solving equations with complex numbers, the first step is to isolate the squared term. This means you want to get the term that includes the square all by itself on one side of the equation. In our example, the equation is given as \( (4 m - 7)^{2} = -27 \).Since the squared term \( (4 m - 7)^{2} \) is already isolated, we can proceed to the next step. Remember:

• Look for the term with the square.
• Move any other terms to the opposite side of the equation by using basic algebra (addition, subtraction, multiplication, or division).

taking the square root of both sides

After isolating the squared term, the next step is to take the square root of both sides of the equation. For our example, we have \( (4 m - 7)^{2} = -27 \).Taking the square root of both sides gives us \( 4 m - 7 = \pm \sqrt{ -27 } \).Since the right side has a negative number under the square root, we will get a complex number. Do not forget the \( \pm \) symbol, as it represents two possible solutions. Always remember:

• Taking the square root of both sides will give you positive and negative roots.
• For negative numbers under the square root, use the imaginary unit \( i \) where \( i^{2} = -1 \).

simplifying complex square roots

With the square root taken, we now simplify it. In our example: \( 4 m - 7 = \pm \sqrt{ -27 } \).To simplify \( \sqrt{ -27 } \), separate it into \( \sqrt{ -1 } \times \sqrt{ 27 } \). Since \( \sqrt{ -1 } = i \), we have \( i \sqrt{ 27 } \). Further breakdown of \( \sqrt{ 27 } \) gives \( 3 \sqrt{ 3 } \). So, \( \sqrt{ -27 } = 3 i \sqrt{ 3 } \). Thus, we can rewrite our equation fully simplified as: 4 m - 7 = \pm 3 i \sqrt{ 3 }. Key points to remember:

• Separate the real and imaginary parts of the square root.
• Simplify the square root of the real number as you normally would.
• Combine the simplified roots back together.

solving for variables

The final step is solving for the variable. We already have two equations from the \( \pm \) sign:

• \( 4 m - 7 = 3 i \sqrt{ 3 } \)
• \( 4 m - 7 = -3 i \sqrt{ 3 } \)

For the first equation: \( 4 m = 3 i \sqrt{ 3 } + 7 \), then \( m = \frac{ 3 i \sqrt{ 3 } + 7 }{4} \). For the second equation: \( 4 m = -3 i \sqrt{ 3 } + 7 \), then \( m = \frac{ -3 i \sqrt{ 3 } + 7 }{4} \). So the solutions are:
\( m = \frac{ 3 i \sqrt{ 3 } + 7 }{ 4 } \) and \( m = \frac{ -3 i \sqrt{ 3 } + 7 } {4 } \). When solving for variables, keep the following in mind:

• Treat each potential solution separately.
• Ensure all terms are correctly simplified and combined.