Problem 65 - College Algebra with Modeling and Visualization

Question

Solve. Write answers in standard form. $$ x^{2}=3 x-5 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{3}{2} + \frac{i\sqrt{11}}{2}\) and \(x = \frac{3}{2} - \frac{i\sqrt{11}}{2}\) as complex numbers.

Step 1: Bring All Terms to One Side

To solve the quadratic equation, first, bring all terms to one side of the equation. Start with the given equation: \( x^2 = 3x - 5 \). To bring all terms to one side, subtract \(3x\) and add \(5\) to both sides, forming \(x^2 - 3x + 5 = 0\).

Step 2: Use the Quadratic Formula

Since the equation \(x^2 - 3x + 5 = 0\) is in standard form \(ax^2 + bx + c = 0\), identify the coefficients: \(a = 1\), \(b = -3\), and \(c = 5\). Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\). Substitute the values: \((-3)^2 - 4 \cdot 1 \cdot 5 = 9 - 20 = -11\). The discriminant is \(-11\).

Step 4: Solve Using the Quadratic Formula

Substitute the values into the quadratic formula: \(x = \frac{-(-3) \pm \sqrt{-11}}{2 \cdot 1}\). Simplify: \(x = \frac{3 \pm \sqrt{-11}}{2}\). Since the discriminant is negative, \(x\) has complex solutions: \(x = \frac{3 \pm i\sqrt{11}}{2}\).

Step 5: Write Answer in Standard Form

Write the solutions in standard form as \(\frac{3}{2} + \frac{i\sqrt{11}}{2}\) and \(\frac{3}{2} - \frac{i\sqrt{11}}{2}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It provides the solution for \(x\) using the formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here is how it works in detail:

• **Identify the coefficients:** These are \(a\), \(b\), and \(c\) from the standard form of the quadratic equation. In our example, \(a = 1\), \(b = -3\), and \(c = 5\).
• **Calculate the discriminant:** This is the part under the square root, \(b^2 - 4ac\). It tells us the nature of the roots. A positive discriminant means two real and distinct roots, zero means one real root, and negative means two complex roots.
• **Use the formula:** Substitute the coefficients into the quadratic formula to find the values of \(x\).

Mastering the quadratic formula will allow you to solve any quadratic equation efficiently.

Standard Form

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). It is essential to convert any given quadratic equation to this form to easily use methods like factorization or the quadratic formula.
This form helps clearly identify coefficients \(a\), \(b\), and \(c\), which can be plugged into the quadratic formula. Here’s how you can convert a given equation into standard form:

• **Move all terms to one side of the equation:** Start by arranging the equation's terms so that all are on one side, resulting in a zero on the other side.
• **Rearrange the terms:** Make sure the terms are in descending order according to their powers, \(x^2\), \(x\), and then the constant term.

Using the example \(x^2 = 3x - 5\), you would rearrange it to \(x^2 - 3x + 5 = 0\). This step is crucial as it sets the stage for straightforward application of solution methods like the quadratic formula.

Complex Numbers

When solving quadratic equations, you might encounter complex numbers, especially when the discriminant \(b^2 - 4ac\) is negative. Complex numbers extend the real number system by introducing an imaginary unit \(i\), defined as \(\sqrt{-1}\).
For the equation \(x^2 - 3x + 5 = 0\), the discriminant is \(-11\), so the solutions are complex. Here’s a quick guide on dealing with complex numbers:

• **Imaginary unit (\(i\)):** Defined as \(\sqrt{-1}\), any square root of a negative number can be expressed using \(i\). For example, \(\sqrt{-11} = i\sqrt{11}\).
• **Express in standard form:** Solutions with complex numbers often appear as \(a + bi\). For our equation, this results in the solutions \(\frac{3}{2} + \frac{i\sqrt{11}}{2}\) and \(\frac{3}{2} - \frac{i\sqrt{11}}{2}\).

Understanding complex numbers allows you to solve quadratic equations, even when real number solutions do not exist.