Problem 52 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution is \( x = 0.900 \).

Step 1: Identify the coefficients

The quadratic equation is given as \( x^2 - 1.800x + 0.810 = 0 \). Identify the coefficients: \( a = 1 \), \( b = -1.800 \), and \( c = 0.810 \). These are the constants you will use in the quadratic formula.

Step 2: Write the quadratic formula

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

Step 3: Calculate the discriminant

Compute the discriminant using the formula \( b^2 - 4ac \). Substitute the values: \( (-1.800)^2 - 4 \times 1 \times 0.810 = 3.24 - 3.24 = 0 \). The discriminant is 0, indicating one real solution.

Step 4: Solve using the quadratic formula

Since the discriminant is 0, use the simplified quadratic formula \( x = \frac{-b}{2a} \). Calculate the roots: \( x = \frac{-(-1.800)}{2 \times 1} = \frac{1.800}{2} = 0.900 \).

Step 5: Verify the solution

Substitute \( x = 0.900 \) back into the original equation to confirm it works: \((0.900)^2 - 1.800 \times 0.900 + 0.810 = 0.810 - 1.620 + 0.810 = 0 \). This confirms that our solution is correct.

Key Concepts

Discriminant Calculation

When working with quadratic equations, the discriminant helps us understand the nature of the solutions. The discriminant is found within the quadratic formula, and its value determines how many and what type of solutions a quadratic equation has. To calculate it, you use the formula: \( b^2 - 4ac \). Here, \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).
In our example, we identified \( a = 1 \), \( b = -1.800 \), and \( c = 0.810 \). Plugging these into our discriminant formula gives us:

• \( (-1.800)^2 = 3.24 \)
• \( 4 \cdot 1 \cdot 0.810 = 3.24 \)

This results in a discriminant value of \( 3.24 - 3.24 = 0 \).
A discriminant of 0 tells us that there is exactly one real solution, which is repeated or doubly counted in different contexts. This is thanks to the fact that the square root of zero doesn’t affect the formula.

Real Solutions

Real solutions of a quadratic equation are found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). The real solutions refer to the x-values where the quadratic equation crosses the x-axis.
The discriminant value significantly influences these real solutions:

• If the discriminant is greater than zero, there are two distinct real solutions.
• If the discriminant is exactly zero, there is one real repeated solution. This is what occurs in our current example, \( 0 \).
• If the discriminant is less than zero, there are no real solutions. Instead, the solutions are complex or imaginary numbers.

Understanding these cases is crucial to predicting how many times a parabola will intersect the x-axis or if it will be displaced above or below the axis entirely.

Quadratic Equation

The quadratic equation is a second-degree polynomial represented in the general form: \( ax^2 + bx + c = 0 \). Solving these equations is a common task in algebra and they usually result in two solutions for \( x \), thanks to the power of two present in the equation.
In solving the quadratic equation, following through with the quadratic formula process is often necessary. This involves:

• Identifying the coefficients \( a \), \( b \), and \( c \).
• Calculating the discriminant, \( b^2 - 4ac \).
• Applying the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

These steps reveal the roots which could be either both real, one real (as in our example), or both complex. Quadratics are fundamental in describing many phenomena in sciences, economics, and engineering, forming the basis for parabolic graphs.