Problem 114 - Algebra for College Students

Question

For each equation, find approximate solutions rounded to two decimal places. $$x^{2}-1.3 x=22.3-x^{2}$$

Step-by-Step Solution

Answer

x ≈ 3.69 or -3.03

Step 1: Move all terms to one side of the equation

First, rewrite the equation so that all terms are on one side. Start with the given equation: \(x^{2} - 1.3x = 22.3 - x^{2}\) Add \(x^{2}\) to both sides to combine like terms: \(x^{2} + x^{2} - 1.3x = 22.3\) This simplifies to: \(2x^{2} - 1.3x - 22.3 = 0\)

Step 2: Apply the quadratic formula

The quadratic formula \(x = \frac{-b \,\pm \, \sqrt{b^{2}-4ac}}{2a}\) is used to solve quadratic equations of the form \(ax^{2} + bx + c = 0\). For the equation \(2x^{2} - 1.3x - 22.3 = 0\): \(a = 2\), \(b = -1.3\), and \(c = -22.3\). Substitute these values into the formula.

Step 3: Substitute the coefficients into the quadratic formula

Substitute \(a\), \(b\), and \(c\) into the quadratic formula: \(x = \frac{-(-1.3) \,\pm \, \sqrt{(-1.3)^{2} - 4 \,\cdot \, 2 \,\cdot \, (-22.3)}}{2 \,\cdot \, 2}\) This simplifies to: \(x = \frac{1.3 \,\pm \, \sqrt{1.69 + 178.4}}{4}\) This simplifies further to: \(x = \frac{1.3 \,\pm \, \sqrt{180.09}}{4}\)

Step 4: Calculate the square root and solve for x

Calculate the square root: \(\sqrt{180.09} \approx 13.42\) Substitute back into the equation: \(x = \frac{1.3 \, \pm \, 13.42}{4}\) This gives two solutions: \(x_{1} = \frac{1.3 + 13.42}{4} \approx 3.69\) \(x_{2} = \frac{1.3 - 13.42}{4} \approx -3.03\)

Key Concepts

Understanding the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation is any equation that can be written in the form \(ax^{2} + bx + c = 0\). It's called 'quadratic' because it contains at least one term that is squared (the \(x^{2}\) term).
The quadratic formula is given by: \[x = \frac{-b \,\pm\, \sqrt{b^{2}-4ac}}{2a}\] This formula might look complicated at first, but it's just a method to find the values of x that make the equation true.
To use the quadratic formula, you need three numbers (called coefficients) from the quadratic equation: \(a\), \(b\), and \(c\). These coefficients correspond to the terms \(ax^{2}\), \(bx\), and \(c\) in the equation.

Breaking Down a Quadratic Equation

In the equation \(x^{2} - 1.3x = 22.3 - x^{2}\), the first step involves rewriting it so that all terms are on one side. This transforms the equation into the standard form \(ax^{2} + bx + c = 0\).
Start with the given equation: \(x^{2} - 1.3x = 22.3 - x^{2}\)
Add \(x^{2}\) to both sides to combine like terms: \(x^{2} + x^{2} - 1.3x = 22.3\)
This results in: \(2x^{2} - 1.3x - 22.3 = 0\)
Now, the equation is in the standard form with \(a = 2\), \(b = -1.3\), and \(c = -22.3\). These values can be used in the quadratic formula to find the solutions for x.

Applying Algebra Steps to Solve the Equation

Once the quadratic equation is in the standard form, you can use algebra steps and the quadratic formula to find the solutions. Here are the steps broken down:

• Identify coefficients \(a\), \(b\), and \(c\) from the equation: \(2x^{2} - 1.3x - 22.3 = 0\) gives \(a = 2\), \(b = -1.3\), \(c = -22.3\).
• Substitute these values into the quadratic formula: \[x = \frac{-(-1.3) \, \pm \, \sqrt{(-1.3)^{2} - 4 \, \cdot \, 2 \, \cdot \, (-22.3)}}{2 \, \cdot \, 2}\]
• This simplifies to: \[x = \frac{1.3 \, \pm \, \sqrt{1.69 + 178.4}}{4}\]
• Then it further simplifies to: \[x = \frac{1.3 \, \pm \, \sqrt{180.09}}{4}\]

Now calculate the square root and solve for x:
\(\text{\sqrt{180.09} \approx 13.42}\)
Substitute back: \[x = \frac{1.3 \, \pm \, 13.42}{4}\]
This gives two solutions:

• \(x_{1} = \frac{1.3 + 13.42}{4} \approx 3.69\)
• \(x_{2} = \frac{1.3 - 13.42}{4} \approx -3.03\)

These are the approximate solutions to the original quadratic equation.