Problem 56 - Algebra for College Students

Question

Expressing solutions to the nearest one-thousandth. $$x^{2}+6 x-44=0$$

Step-by-Step Solution

Answer

The solutions are approximately 4.280 and -10.280.

Step 1: Write the Quadratic Formula

The quadratic formula is used to find the solutions (roots) of a quadratic equation given in the standard form \(ax^2 + bx + c = 0\). It is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). For this equation, \(a = 1\), \(b = 6\), and \(c = -44\).

Step 2: Calculate the Discriminant

The discriminant \(b^2 - 4ac\) determines the nature of the roots. Compute it using the values we have: \(b^2 - 4ac = 6^2 - 4 \times 1 \times (-44) = 36 + 176 = 212\).

Step 3: Apply the Quadratic Formula

With the discriminant calculated, substitute it into the quadratic formula: \(x = \frac{-6 \pm \sqrt{212}}{2 \times 1}\).

Step 4: Simplify the Expression

Approximate \(\sqrt{212}\). We find that \(\sqrt{212} \approx 14.560\). Now substitute this back into the equation: \(x = \frac{-6 \pm 14.560}{2}\).

Step 5: Calculate the Solutions

Calculate the two possible solutions from \(x = \frac{-6 + 14.560}{2}\) and \(x = \frac{-6 - 14.560}{2}\). This gives \(x_1 \approx 4.280\) and \(x_2 \approx -10.280\).

Step 6: Round to the Nearest Thousandth

Round each solution to the nearest thousandth. Thus, \(x_1 \) remains \(4.280\) and \(x_2 \) remains \(-10.280\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation, an equation of the form \(ax^2 + bx + c = 0\). To recall, the formula is written as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula is universal, meaning it can solve any quadratic equation, as long as the coefficients \(a\), \(b\), and \(c\) are real numbers. To apply the quadratic formula properly, it's essential to first organize your equation such that it is in the standard form. Next, identify terms \(a\), \(b\), and \(c\) and substitute them into the formula. The formula involves calculating a part known as the discriminant, which we will discuss further. The \(\pm\) sign indicates there are typically two possible solutions, stemming from addition and subtraction of the square root term.

Discriminant

The discriminant is a crucial component of the quadratic formula found within it as \(b^2 - 4ac\). It helps to determine the nature and number of solutions a quadratic equation might have.- **Positive Discriminant**: If \(b^2 - 4ac > 0\), there are two distinct real roots. This means the parabola crosses the x-axis at two points.- **Zero Discriminant**: If \(b^2 - 4ac = 0\), there is exactly one real root. This is when the parabola just touches the x-axis (called a repeated or double root).- **Negative Discriminant**: If \(b^2 - 4ac < 0\), there are no real roots; instead, the solutions are complex or imaginary.In our problem, the discriminant \(b^2 - 4ac\) was found to be 212, which is positive, indicating the existence of two distinct real roots. This information is crucial before progressing to compute the exact roots.

Rounding to Thousandths

Once the two solutions are found using the quadratic formula, they can often be non-integers. For practical applications or simpler expressions, we round numbers to a specified degree of precision. In mathematical problems, "rounding to the nearest thousandth" means keeping three decimal places.To round a number to the nearest thousandth:- Look at the fourth decimal place.- If it is 5 or greater, increase the third decimal place by one.- If it is less than 5, keep the third decimal place the same.In our solution \(x_1 \approx 4.280\) and \(x_2 \approx -10.280\), already rounded to thousandths, maintained their values since the digits following the thousandth place did not meet the criteria for rounding up. Rounding ensures precision while simplifying expressions for easier use and understanding.