Problem 59 - Algebra for College Students

Question

Expressing solutions to the nearest one-thousandth. $$x^{2}+9 x+3=0$$

Step-by-Step Solution

Answer

The solutions are approximately \(x_1 = -0.347\) and \(x_2 = -8.653\).

Step 1: Identify the coefficients

The given quadratic equation is \(x^{2} + 9x + 3 = 0\). Identify the coefficients: - \(a = 1\) (the coefficient of \(x^2\)),- \(b = 9\) (the coefficient of \(x\)),- \(c = 3\) (the constant term).

Step 2: Write the quadratic formula

The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Use this formula to solve the equation \(x^{2} + 9x + 3 = 0\).

Step 3: Calculate the discriminant

The discriminant \(\Delta\) is calculated as \(b^2 - 4ac\). Substitute the known values:\[\Delta = 9^2 - 4 \cdot 1 \cdot 3 = 81 - 12 = 69\]

Step 4: Substitute values into the quadratic formula

Now substitute the coefficients and the discriminant into the quadratic formula:\[x = \frac{-9 \pm \sqrt{69}}{2 \cdot 1}\]This simplifies to:\[x = \frac{-9 \pm \sqrt{69}}{2}\]

Step 5: Calculate the square root

Find the square root of the discriminant (69) using a calculator:\[\sqrt{69} \approx 8.3066\]

Step 6: Calculate the solutions

Substitute \(\sqrt{69} \approx 8.3066\) back into the solutions:\[x_1 = \frac{-9 + 8.3066}{2} \approx \frac{-0.6934}{2} = -0.3467\]\[x_2 = \frac{-9 - 8.3066}{2} \approx \frac{-17.3066}{2} = -8.6533\]

Step 7: Round to the nearest one-thousandth

Round the solutions from Step 6 to the nearest one-thousandth. Thus:\[x_1 \approx -0.347\]\[x_2 \approx -8.653\]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). By substituting the coefficients \(a\), \(b\), and \(c\) into the formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \], you can find the solutions (or roots). This formula helps us solve even when the roots are not easily factorable. It's important to correctly identify your coefficients from the initial equation to use this formula accurately.

Discriminant

Understanding the discriminant is crucial since it informs us about the nature of the roots without solving the entire equation. The discriminant is part of the quadratic formula under the radical sign: \(b^2 - 4ac\). Depending on its value, it can indicate different types of solutions:

• If \(\Delta > 0\), there are two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root (a repeated root).
• If \(\Delta < 0\), there are no real roots, but two complex roots.

For the equation \(x^2 + 9x + 3 = 0\), the discriminant was found to be 69, showing two distinct real roots.

Root Approximation

When the square root of the discriminant doesn't yield a neat number, root approximation becomes necessary. Calculators or estimation methods help to approximate values, making it easier to write simplified solutions. With our equation, \(\sqrt{69} \approx 8.3066\). Using this approximation in further steps aids in developing precise answers. Always remember to follow through on your calculations to maintain exactness.

Rounding Numbers

Rounding is the process of simplifying a number to make it easier to work with. In mathematical solutions, especially involving irrational numbers, rounding to a specified degree of precision can be vital. In our example, solutions were rounded to the nearest one-thousandth. This step means taking the approximate solutions, like \(-0.3467\), and rounding it to \(-0.347\). The necessity of rounding ensures that answers are easy to read and consistent, while still being as accurate as possible given the context.