Problem 48 - Precalculus Student Solutions Manual 5th

Question

Solve each equation by completing the square. $$-3 x^{2}+9 x=7$$

Step-by-Step Solution

Answer

The solutions are \[x = \frac{3}{2} + \frac{\text{i}}{\text{2}\text{sqrt{3}}}\] and \[x = \frac{3}{2}-\frac{\text{i}}{\text{2}\text{sqrt{3}}}\].

Step 1: Move the constant term to the other side

Start by moving the constant term on the right side of the equation: \[-3x^2 + 9x = 7 \rightarrow -3x^2 + 9x - 7 = 0\].

Step 2: Divide through by the coefficient of the quadratic term

Divide every term by \[-3\] to make the coefficient of the quadratic term equals to 1: \[x^2 - 3x = -\frac{7}{3}\].

Step 3: Complete the square

To complete the square, add and subtract \[(\frac{-3}{2})^2 = \frac{9}{4}\] to the left side of the equation: \[x^2 - 3x + \frac{9}{4} = -\frac{7}{3} + \frac{9}{4}\].

Step 4: Simplify the equation

Combine the fractions on the right side: \[-\frac{7}{3} + \frac{9}{4} = -\frac{28}{12} + \frac{27}{12} = -\frac{1}{12}\]. So now the equation is: \[(x - \frac{3}{2})^2 = -\frac{1}{12}\].

Step 5: Solve for x

Take the square root of both sides: \[x - \frac{3}{2} = \frac{\text{i}}{\text{2}\text{sqrt{3}}}\] or \[x - \frac{3}{2} = -\frac{\text{i}}{\text{2}\text{sqrt{3}}}\]. Therefore, the solutions are: \[x = \frac{3}{2} + \frac{\text{i}}{\text{2}\text{sqrt{3}}}\] and \[x = \frac{3}{2} - \frac{\text{i}}{\text{2}\text{sqrt{3}\].

Key Concepts

quadratic equations

Quadratic equations are mathematical expressions of the form ax² + bx + c = 0. Here, 'a', 'b', and 'c' are constants, and 'x' represents the variable. Quadratic equations often appear in various fields such as physics, engineering, and economics. There are several methods for solving quadratic equations:

• Factoring
• Using the quadratic formula
• Graphing
• Completing the square

In this exercise, we used the technique of completing the square. This method involves manipulating the equation into a perfect square trinomial, which can then be solved by taking the square root of both sides.

algebraic techniques

Algebraic techniques refer to the set of operations and rules used to manipulate algebraic expressions and equations. In our example, we used several key algebraic techniques:

• Moving the constant term
• Dividing by the coefficient of the quadratic term
• Completing the square
• Combining fractions

Moving the constant term is done by adding or subtracting the constant from both sides of the equation. This isolates terms involving the variable on one side.

Dividing by the coefficient is another common technique. For example, we divided by -3 to make the coefficient of x² equal to 1. Completing the square involves adding and subtracting the same value to make the left-hand side a perfect square trinomial. Finally, combining fractions is a necessary skill for simplifying terms after completing the square.

complex solutions

In the given exercise, after completing the square, we encountered complex solutions. Complex solutions arise when taking the square root of a negative number. These solutions involve the imaginary unit, denoted as 'i', which is defined by the equation i² = -1.

After completing the square, we had: \[(x - \frac{3}{2})² = -\frac{1}{12}\]. Taking the square root of both sides, we obtained: \[x - \frac{3}{2} = \frac{i}{2\sqrt{3}}\] or \[x - \frac{3}{2} = -\frac{i}{2\sqrt{3}}\]. Complex solutions often occur in quadratic equations when the discriminant (the part under the square root in the quadratic formula) is less than zero. Complex numbers provide a more complete understanding of the behavior and solutions of quadratic equations, especially in fields like electrical engineering and quantum physics.