Problem 47 - Brief Applied Calculus

Question

Solve each equation by factoring or the Quadratic Formula, as appropriate. $$ -4 x^{2}+12 x=8 $$

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = 2\).

Step 1: Move all terms to one side

Rearrange the equation so it is set to zero. Start with \(-4x^2 + 12x = 8\). Move 8 to the other side by subtracting 8 from both sides:\(-4x^2 + 12x - 8 = 0\).

Step 2: Factor out the greatest common factor

Identify the greatest common factor for the terms on the left side. The terms \(-4x^2 + 12x - 8\) can be divided by -4:\(-4(x^2 - 3x + 2) = 0\).

Step 3: Factor the quadratic expression

Now we need to factor the quadratic expression \(x^2 - 3x + 2\). Look for two numbers that multiply to 2 and add to -3. The numbers -1 and -2 fit:\((x - 1)(x - 2) = 0\).

Step 4: Use the Zero Product Property

Set each factor in the equation \((x - 1)(x - 2) = 0\) equal to zero:First factor: \(x - 1 = 0\) gives \(x = 1\).Second factor: \(x - 2 = 0\) gives \(x = 2\).

Step 5: Verify the solutions

Substitute \(x = 1\) and \(x = 2\) back into the original equation \(-4x^2 + 12x - 8 = 0\) to ensure both satisfy the equation.For \(x = 1\): \(-4(1)^2 + 12(1) - 8 = -4 + 12 - 8 = 0\).For \(x = 2\): \(-4(2)^2 + 12(2) - 8 = -16 + 24 - 8 = 0\).Both solutions satisfy the original equation.

Key Concepts

Factoring Method

The factoring method is a powerful tool often used to solve quadratic equations. The main idea is to express the quadratic expression as a product of two linear factors. This method relies on the ability to spot the structure of the quadratic and find numbers that meet specific conditions.

When dealing with an equation like displaylatex{blockquote: -4x^2 + 12x - 8 = 0},you start by ensuring the equation is set to zero. Then, look for the greatest common factor, which in this case simplifies the quadratic to -4 multiplied by a simpler expression. This lets you focus on factoring the remaining quadratic, displaylatex{blockquote: (x^2 - 3x + 2)}.

To do this, identify two numbers that multiply to the constant term (here, 2) and add to the linear coefficient (-3). In this example, these numbers are -1 and -2. Thus, the expression becomes:\[ (x - 1)(x - 2) = 0 \]

This method is efficient for quadratics that can be neatly factored into integers. However, some quadratics might require other methods if they do not factor easily.

Quadratic Formula

When the factoring method doesn't neatly work, the Quadratic Formula is a reliable alternative. This formula can solve any quadratic equation, providing invaluable flexibility. The formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a\), \(b\), and \(c\) are the coefficients of the terms in any quadratic equation of the form \(ax^2 + bx + c = 0\).

The Quadratic Formula gives two possible solutions, due to the 'plus or minus' (functionendpoint{pm}) in the formula, which accounts for the two roots of the quadratic equation.

In any given scenario where factoring proves difficult, such as when roots are irrational or complex, the Quadratic Formula will lead you to a robust solution.

The formula utilizes the discriminant, \(b^2 - 4ac\), which provides insight into the nature of the roots (real or complex). A positive discriminant indicates distinct real roots, while zero points to a double root, and a negative value suggests complex roots.

Zero Product Property

The Zero Product Property is a pivotal concept in solving quadratic equations by factoring. It states that if the product of two numbers is zero, then at least one of the factors must be zero. This principle is written simply as: if \(ab = 0\), then either \(a = 0\) or \(b = 0\).

In the example equation, displaylatex{blockquote: (x - 1)(x - 2) = 0},applying the Zero Product Property allows you to break the problem into smaller parts by setting each factor equal to zero:

• \(x - 1 = 0\), which gives \(x = 1\)
• \(x - 2 = 0\), which gives \(x = 2\)

The Zero Product Property is straightforward yet incredibly effective, simplifying how we determine potential solutions from factored expressions. It ensures that solving these types of equations can be systematic and logical, always leading to the correct roots.