Problem 39 - Brief 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 \(x = 4\) and \(x = 5\).

Step 1: Rearrange the Equation

Start by moving all terms to one side of the equation so that it is in the standard quadratic form \(ax^2 + bx + c = 0\). \[2x^2 + 40 - 18x = 0\]Reorder the terms:\[2x^2 - 18x + 40 = 0\]

Step 2: Simplify the Equation

Notice that each term in the equation can be divided by 2 for simplification: \[x^2 - 9x + 20 = 0\]

Step 3: Factor the Quadratic Equation

We need to factor the quadratic equation \(x^2 - 9x + 20 = 0\). Look for two numbers that multiply to 20 and add to -9. The numbers are -4 and -5:\[(x - 4)(x - 5) = 0\]

Step 4: Solve for x

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

Step 5: Verify the Solutions

Substitute \(x = 4\) and \(x = 5\) back into the original equation to ensure they satisfy it.1. For \(x = 4\), \(2(4)^2 + 40 = 18(4)\) gives 72 = 72.2. For \(x = 5\), \(2(5)^2 + 40 = 18(5)\) gives 90 = 90.Both values are correct.

Key Concepts

Factoring

Factoring is a method used to solve quadratic equations, and is often preferred when the equation can be easily rewritten as a product of linear factors. A quadratic equation like \[ x^2 - 9x + 20 = 0 \]can be factored by finding two numbers that multiply to the constant term (in this case, 20) and add up to the coefficient of the linear term (in this case, -9).

• Look for two numbers whose product is 20 and sum is -9, which are -4 and -5.
• Thus, the equation can be written as \[(x - 4)(x - 5) = 0\]

By setting each factor equal to zero, we can solve for the values of \(x\) that satisfy the equation. Factoring is efficient and reveals potential solutions quickly, provided that finding suitable factors is straightforward.

Quadratic Formula

The Quadratic Formula is another powerful tool for solving quadratic equations and can be used when factoring seems difficult or cumbersome. It's expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation in standard form \(ax^2 + bx + c = 0\).Utilizing the Quadratic Formula allows you to:

• Solve any quadratic equation, even when it cannot be easily factored.
• Determine the roots' nature (real or complex) from the discriminant, \(b^2 - 4ac\).

In our specific example though, factoring was simpler than using the formula, so it was the method of choice.

Problem Solving Steps

Problem-solving with quadratic equations typically involves several hallmark steps. Starting by rewriting the equation in standard form is crucial:

• Move all terms to one side so it looks like \(ax^2 + bx + c = 0\).
• Simplify the equation if possible to ease further operations.
• Choose a method to solve the equation, like factoring or the quadratic formula.
• Verify the solution by reinserting the values back into the original equation.

These steps ensure a systematic approach that clarifies the solving process from start to finish, enhancing understanding and result accuracy.

Equation Simplification

Simplifying the equation often makes the ensuing steps more manageable and less error-prone. During simplification, you aim to make the coefficients as small as possible, which can be achieved through:

• Identifying common factors in the terms.
• Dividing through by these factors before attempting further solution methods.

In our equation,\[2x^2 - 18x + 40 = 0\]dividing by 2 yields the simpler form\[x^2 - 9x + 20 = 0\],allowing for easy identification of factors. Simplification can reveal hidden patterns that make solving by factoring or using formulas more approachable.