Problem 50 - Brief Applied Calculus

Question

Solve each equation by factoring or the Quadratic Formula, as appropriate. $$ 2 x^{2}-8 x+10=0 $$

Step-by-Step Solution

Answer

The solutions are complex: \( x = 2 + i \) and \( x = 2 - i \).

Step 1: Identify the Equation Type

The given equation is a quadratic equation because it is in the form \( ax^2 + bx + c = 0 \), where \( a = 2 \), \( b = -8 \), and \( c = 10 \).

Step 2: Simplify the Equation

To make factoring easier, first check if the coefficients have a common factor. In this case, we can divide the entire equation by 2: \( x^2 - 4x + 5 = 0 \).

Step 3: Check for Factorability

Look for two numbers that multiply to \( ac = 1 \times 5 = 5 \) and add up to \( b = -4 \). Since no such integers exist, the equation is not easily factorable.

Step 4: Apply the Quadratic Formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve the equation. Here, \( a = 1 \), \( b = -4 \), and \( c = 5 \).

Step 5: Calculate the Discriminant

Calculate \( b^2 - 4ac = (-4)^2 - 4 \times 1 \times 5 = 16 - 20 = -4 \). The negative discriminant indicates that the solutions are complex.

Step 6: Find the Roots

Substitute the values into the quadratic formula: \( x = \frac{-(-4) \pm \sqrt{-4}}{2 \times 1} \). Simplify to find the solutions: \( x = \frac{4 \pm 2i}{2} \), resulting in \( x = 2 + i \) and \( x = 2 - i \).

Key Concepts

Factoring

Factoring is a method used to simplify quadratic equations, making them easier to solve. The idea is to express the quadratic equation as a product of two binomials. For example, the equation \( ax^2 + bx + c = 0 \) can sometimes be written as \((px + q)(rx + s) = 0\). By setting each binomial equal to zero, you can find the solutions or roots of the equation.

• First, identify if there is a common factor in all terms. This can simplify the equation.
• Next, find two numbers that multiply to \(ac\) and add to \(b\). These numbers help break down the middle term if factored directly.

When a quadratic equation is not easily factorable, other methods, like the Quadratic Formula, come into play. As seen in the original exercise example \(2x^2 - 8x + 10 = 0\), after simplifying, the equation \(x^2 - 4x + 5 = 0\) isn’t easily factored because there are no rational numbers multiplying to 5 and adding to -4. Hence, factoring isn’t always the best route.
In this exercise, the Quadratic Formula becomes a necessary tool.

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation, even if it isn't easily factorable. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), and the quadratic formula provides the solutions:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Here's how to apply it:

• Identify coefficients \(a\), \(b\), and \(c\) from the equation.
• Calculate the discriminant \(b^2 - 4ac\). This value shows the nature of the roots.
• Substitute these values into the formula.

In our example \(x^2 - 4x + 5 = 0\), the discriminant \(-4\) was found to be negative, leading us to complex solutions. The quadratic formula is versatile and guarantees a solution even if the roots are not real numbers. This reliability makes it a fundamental method in algebra for solving quadratic equations.

Complex Solutions

Complex solutions arise when the discriminant \((b^2 - 4ac)\) of a quadratic equation is negative. This indicates that the roots of the equation are not real numbers but are complex. Complex numbers include a real part and an imaginary part, typically written in the form \(a + bi\), where \(i\) is the square root of -1.
In our solved equation, the discriminant is \(-4\), leading to the solutions \(2 + i\) and \(2 - i\). Calculating complex solutions:

• The discriminant gives a negative inside the square root in the quadratic formula.
• Use \(i\) for \(\sqrt{-1}\) to handle negative square roots.
• Simplify the found expression to standard complex form.

Having complex solutions indicates that the graph of the quadratic equation does not intersect the x-axis, affirming that no real roots exist. Understanding complex solutions allows students to grasp that not all equations have solutions through real numbers, expanding their skill set for handling various types of mathematical problems.