Problem 56 - Algebra and Trigonometry

Question

Use the quadratic formula and a calculator to find all real solutions, rounded to three decimals. $$ x^{2}-1.800 x+0.810=0 $$

Step-by-Step Solution

Answer

The real solution is \( x = 0.900 \).

Step 1: Identify coefficients

The given quadratic equation is \( x^2 - 1.800x + 0.810 = 0 \). Identify the coefficients as follows: \( a = 1 \), \( b = -1.800 \), and \( c = 0.810 \).

Step 2: Write the quadratic formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We will use this formula to find the roots of the quadratic equation.

Step 3: Calculate the discriminant

Use the formula for the discriminant, \( b^2 - 4ac \). Here, \( b = -1.800 \), \( a = 1 \), and \( c = 0.810 \). So, \( b^2 = (-1.800)^2 = 3.240 \) and \( 4ac = 4 \times 1 \times 0.810 = 3.240 \). The discriminant is \( 3.240 - 3.240 = 0 \).

Step 4: Solve using the quadratic formula

Since the discriminant is 0, there is one real solution (both roots are the same). Substitute the values into the quadratic formula: \( x = \frac{-(-1.800) \pm \sqrt{0}}{2 \times 1} \). This simplifies to \( x = \frac{1.800}{2} = 0.900 \).

Step 5: Perform calculation to find the solution

Compute the value \( x = 0.900 \) using a calculator and verify. Since the discriminant was zero, the solution is repeated and only one real, distinct root exists.

Key Concepts

Quadratic Equation

To understand quadratic equations, it's helpful to start with their general form. A quadratic equation is any equation of the form \( ax^2 + bx + c = 0 \). It is called "quadratic" because "quad" means square, and the variable \( x \) in the equation is squared. Here is what each part represents:

• \( a \): the coefficient of \( x^2 \).
• \( b \): the coefficient of \( x \).
• \( c \): the constant term.

Quadratic equations are fundamental in algebra and appear in various real-world applications like physics, engineering, and economics. To solve these equations, we often use methods such as factoring, completing the square, or the quadratic formula. Each method is useful in different situations, but the quadratic formula is powerful because it works for any quadratic equation.

Discriminant

The discriminant is a key part of the quadratic formula, and it tells us valuable information about the nature of the roots of a quadratic equation. It is found within the square root part of the formula: \( b^2 - 4ac \).
The discriminant allows us to determine:

• **If the roots are real or complex**: If the discriminant is positive, there are two distinct real roots. If it is zero, there is exactly one real root (a repeated or double root). If negative, the roots are complex or imaginary.
• **How the roots behave**: Positive discriminants imply distinct roots, while zero discriminants indicate they are identical.

In our exercise, we calculated \( b^2 - 4ac = 0 \), so we know there is exactly one real solution. Understanding the discriminant can quickly guide us in solving quadratic equations, even before performing the detailed solutions.

Real Solutions

Real solutions are the values of \( x \) that satisfy the quadratic equation, meaning they make the equation true. These solutions are drawn from the possible real numbers rather than imaginary or complex numbers.
When solving a quadratic, we are interested in finding these real values of \( x \), which occur at the points where the parabola crosses the x-axis in a graph of the function \( f(x) = ax^2 + bx + c \).

• If the quadratic equation has real solutions, the graph will intersect the x-axis. The number of intersections will match the number of real roots discovered.
• In this exercise, we found one real solution, \( x = 0.900 \), because the discriminant was zero, leading us to a double root, or a repeated solution.

Grasping these concepts helps in visualizing and solving quadratic equations more efficiently.

Coefficients

Coefficients are the numerical factors associated with the variables in a quadratic equation. They are essential in manipulating and solving such equations.

• The coefficient \( a \) determines the "width" and direction of the parabola's opening in the graph of the quadratic. If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards.
• The coefficient \( b \) influences the slope or tilt of the parabola.
• The constant \( c \) affects the vertical shift of the parabola, essentially moving it up or down the y-axis.

In our exercise, the coefficients are \( a = 1 \), \( b = -1.800 \), and \( c = 0.810 \). Understanding these helps in plugging values into the quadratic formula and predicting the nature of the solutions.