Problem 39 - Applied Calculus

Question

Solve each equation by factoring or the Quadratic Formula, as appropriate. $$ 2 x^{2}+40=18 x $$

Step-by-Step Solution

Answer

The solutions are \( x = 5 \) and \( x = 4 \).

Step 1: Write the Equation in Standard Form

Start by moving all terms to one side of the equation to set it to zero. Begin with the equation: \[ 2x^2 + 40 = 18x \] Subtract \( 18x \) from both sides to achieve this: \[ 2x^2 - 18x + 40 = 0 \]

Step 2: Simplify the Equation

To simplify the equation, divide all terms by 2 to make the coefficients smaller and easier to work with: \[ x^2 - 9x + 20 = 0 \]

Step 3: Factor the Quadratic Equation

Now, we attempt to factor the quadratic. Look for two numbers whose product is 20 and sum is -9. These numbers are -5 and -4. Hence, the equation factors as: \[ (x - 5)(x - 4) = 0 \]

Step 4: Solve for the Variable

Set each factor equal to zero and solve for \( x \): 1. \( x - 5 = 0 \) which gives \( x = 5 \) 2. \( x - 4 = 0 \) which gives \( x = 4 \)

Step 5: Conclusion

The solutions to the equation are \( x = 5 \) and \( x = 4 \).

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to solve quadratic equations by expressing them as products of linear factors. For a quadratic equation in the form \( ax^2 + bx + c = 0 \), the goal is to find two numbers that multiply to \( ac \) (the product of the coefficient of \( x^2 \) and the constant term) and add up to \( b \) (the coefficient of \( x \)). This process is essential because it breaks down the quadratic into simpler expressions, allowing you to solve for the variable.

• Step 1: Rearrange the equation to standard form \( ax^2 + bx + c = 0 \).
• Step 2: Identify two numbers that multiply to \( ac \) and add up to \( b \).
• Step 3: Rewrite the middle term using these two numbers, and then factor by grouping.
• Step 4: Express the quadratic as a product of two binomials.

In our example, \( x^2 - 9x + 20 = 0 \), we realized that \(-5\) and \(-4\) are the numbers that satisfy both conditions, leading us to factor the equation into \((x - 5)(x - 4) = 0\). This method provides a structural approach to simplify solving quadratics.

Quadratic Formula

The quadratic formula is a powerful tool that solves all types of quadratic equations, even when they cannot be factored easily. The formula is represented as:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]This formula is derived from the process of completing the square, and it allows you to find the solutions for any quadratic equation \( ax^2 + bx + c = 0 \).

• Step 1: Identify the coefficients \( a \), \( b \), and \( c \) from the equation.
• Step 2: Substitute these values into the quadratic formula.
• Step 3: Calculate the discriminant \( b^2 - 4ac \). If it's non-negative, proceed to evaluate using the formula.
• Step 4: Compute the results using both the plus and minus options in \( \pm \) to find the two solutions for \( x \).

While the quadratic formula is not necessary in our specific example since factoring was sufficient, it is essential for quadratics where simple factor pairs do not exist. The formula ensures a comprehensive method of finding solutions.

Solution of Quadratic Equations

The solution of quadratic equations involves finding the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These values are known as the roots or zeros of the quadratic equation.
There are generally two main techniques to solve these equations: factoring and the quadratic formula. Both methods are used depending on the equation's characteristics and complexity.

• **Factoring** is preferred when the quadratic can be easily expressed as a product of binomials. It provides straightforward solutions, as seen in our example, where the solutions were \( x = 5 \) and \( x = 4 \).
• **Quadratic Formula** is ideal for quadratic equations that do not factor neatly. It uses the coefficients \( a \), \( b \), and \( c \) to calculate solutions directly and is universally applicable.

In conclusion, choosing the right method allows for efficient solving of quadratic equations, whether it’s the simplicity of factoring or the thoroughness of the quadratic formula. Each approach ensures that you can find the roots effectively and with clarity.