Problem 61 - Precalculus Student Solutions Manual 5th

Question

Solve each equation using the quadratic formula. $$0.2 x^{2}+0.4 x-0.3=0$$

Step-by-Step Solution

Answer

The solutions are approximately 0.58125 and -2.58125.

Step 1: Identify coefficients

Identify the coefficients in the quadratic equation 0.2 x^{2}+0.4 x-0.3=0. The coefficients are a=0.2, b=0.4, and c=-0.3.

Step 2: Write the quadratic formula

Recall the quadratic formula: \[x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\] Substitute the coefficients into the formula.

Step 3: Substitute coefficients

Substitute a=0.2, b=0.4, and c=-0.3 into the quadratic formula: \[ x = \frac{-0.4 \pm \sqrt{(0.4)^{2}-4(0.2)(-0.3)}}{2(0.2)}\]

Step 4: Simplify under the square root

Evaluate the expression under the square root: \[ x = \frac{-0.4 \pm \sqrt{0.16+0.24}}{0.4}\] This simplifies to \[ x = \frac{-0.4 \pm \sqrt{0.40}}{0.4}\]

Step 5: Simplify the square root

Evaluate the square root: \[ x = \frac{-0.4 \pm 0.6325}{0.4}\]

Step 6: Solve for x

This gives two possible solutions: \[ x = \frac{-0.4 + 0.6325}{0.4} \approx 0.58125 \]and \[ x = \frac{-0.4 - 0.6325}{0.4} \approx -2.58125 \]

Key Concepts

quadratic equation

A quadratic equation is an equation of the form ewline Example: 0.2x^2 + 0.4x - 0.3 = 0ewline Quadratic equations are essential in algebra. They represent curves known as parabolas when graphed. These curves can open upward or downward. Where

- **a** is the coefficient of x²- **b** is the coefficient of x- **c** is the constant term

coefficients

In a quadratic equation, coefficients are the numerical values that multiply the variables. For instance, in the equation 0.2x² + 0.4x - 0.3 = 0, the coefficients are: - **a** = 0.2- **b** = 0.4- **c** = -0.3 Identifying the coefficients is the first step in solving a quadratic equation using the quadratic formula.

square root simplification

Simplifying the expression under the square root is crucial for solving quadratic equations. This part of the formula, known as the discriminant, determines the nature and number of solutions.ewline For the equation 0.2x^2 + 0.4x - 0.3 = 0, the discriminant is calculated as follows:ewline - Calculate b² - 4ac: [latex and formula to be inserted]0.4^2 - 4(0.2)(-0.3) = 0.16 + 0.24 = 0.40 - Take the square root of the discriminant: Solving this gives √0.40 ≈ 0.6325 This value is then substituted back into the quadratic formula.

solving equations

Once you've simplified under the square root, you can solve for x. This involves plugging values into the quadratic formula and simplifying. For our example: Substitute the values into the formula: ewline x = (-b ± √(b² - 4ac)) / 2a This can be broken down into two potential solutions:ewline - Solve for x when using positive value of the square root: x = (-0.4 + 0.6325) / 0.4 ≈ 0.58125- Solve for x when using negative value of the square root: x = (-0.4 - 0.6325) / 0.4 ≈ -2.58125ewline Both these values are answers of the quadratic equation.