Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, solve the quadratic equation by factoring. $$ 4 x^{2}=5 x $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = \frac{5}{4}\).

Step 1: Write the Equation in Standard Form

Start with the given equation: \(4x^2 = 5x\). To factor, we first need to write the equation in standard form \(ax^2 + bx + c = 0\). Subtract \(5x\) from both sides to get: \(4x^2 - 5x = 0\).

Step 2: Factor Out the Greatest Common Factor

Examine each term to find the greatest common factor (GCF). Both terms, \(4x^2\) and \(5x\), share a common factor of \(x\). Factoring out \(x\) gives us the equation \(x(4x - 5) = 0\).

Step 3: Apply the Zero Product Property

According to the zero product property, if \(a \cdot b = 0\), then \(a = 0\) or \(b = 0\). Apply this property to our equation \(x(4x - 5) = 0\). This gives two separate equations: \(x = 0\) and \(4x - 5 = 0\).

Step 4: Solve Each Equation

First, solve \(x = 0\), which is already given. Next, solve the equation \(4x - 5 = 0\). Add 5 to both sides to get \(4x = 5\). Then divide both sides by 4 to solve for \(x\), giving \(x = \frac{5}{4}\).

Step 5: State All Solutions

Combine the solutions from the two equations. The solutions are \(x = 0\) and \(x = \frac{5}{4}\).

Key Concepts

Factoring Quadratic Equations

Factoring quadratic equations is a method used to solve quadratic equations by expressing them as a product of simpler polynomials. This technique is useful because it transforms a polynomial equation into factors that, when multiplied together, yield the original polynomial. Factoring can simplify the process of finding the roots of a quadratic equation.

To factor a quadratic equation, follow these steps:

• Ensure the equation is in standard form: \(ax^2 + bx + c = 0\).
• Identify terms with similar variables or constants.
• Break down the middle term if needed, to assist with grouping.
• Find pairs of numbers whose product is equal to the constant term \(ac\) and whose sum is equal to the middle coefficient \(b\).
• Rewrite the equation using these numbers to help group terms, then factor by grouping.

Once factored, employ other methods, such as the zero product property, to solve for the variable.

Zero Product Property

The zero product property is a fundamental principle used to solve equations where the product of two factors equals zero. This property states that for any real numbers \(a\) and \(b\), if \(a \cdot b = 0\), then either \(a = 0\) or \(b = 0\) or both. This concept is essential when solving factored equations.

In practice:

• When you have an equation like \(x(4x - 5) = 0\), apply the zero product property.
• Set each factor equal to zero: \(x = 0\) and \(4x - 5 = 0\).
• Solve each resulting simple equation separately to find the possible values of the variable \(x\).
• This approach finds all potential roots or solutions for the original quadratic equation.

Using the zero product property is powerful because it breaks down more complex polynomial expressions into simple solvable components.

Greatest Common Factor (GCF)

Finding the greatest common factor (GCF) is a crucial step when factoring quadratic equations. The GCF is the largest number or expression that divides all terms in a polynomial without remainder. By factoring out the GCF, you simplify the equation, making it easier to solve.

To find the GCF:

• Look at each term in the equation, focusing on coefficients and variables.
• Identify the greatest common factor shared among all terms. In this example, the GCF of \(4x^2\) and \(5x\) is \(x\).
• Factor out the GCF, rewriting the equation to isolate the simplified expression. For example, \(x(4x - 5) = 0\) where \(x\) is the factored GCF.
• Proceed with further steps like using the zero product property to identify solutions.

Factoring out the GCF reduces the complexity, turning potentially challenging equations into simpler ones.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is expressed as \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are coefficients and \(a eq 0\). This form is essential because it provides a structured way to apply various methods like factoring, using the quadratic formula, or completing the square to find the solutions of the equation.

To write an equation in standard form, perform the following:

• Ensure all terms \(ax^2, bx, \, and\, c\) are on the left side of the equation.
• Rearrange terms according to their degree, with \(ax^2\) first, \(bx\) second, and the constant \(c\) last.
• Ensure the equation equals zero.
• This standard form allows easy identification of coefficients for further analysis or solving techniques.

Standard form is a starting point in solving, setting the stage for applying other techniques to find a quadratic's roots.