Problem 75 - Algebra for College Students

Question

Solve each of the equations. $$4 x^{2}=5 x$$

Step-by-Step Solution

Answer

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

Step 1: Set the Equation to Zero

To solve the quadratic equation \(4x^2 = 5x\), we first set the equation to zero by subtracting \(5x\) from both sides. This gives us the equation \(4x^2 - 5x = 0\).

Step 2: Factor the Quadratic

Factor the expression \(4x^2 - 5x\). Begin by factoring out the common factor, which is \(x\), resulting in \(x(4x - 5) = 0\).

Step 3: Solve Each Factor for Zero

According to the zero-product property, if \(ab = 0\) then either \(a = 0\) or \(b = 0\). Apply this property to the factored equation \(x(4x - 5) = 0\). First, set \(x = 0\). Then, set \(4x - 5 = 0\) and solve for \(x\).

Step 4: Solve the Linear Equation

Solve \(4x - 5 = 0\). Add 5 to both sides to get \(4x = 5\). Next, divide both sides by 4 to isolate \(x\), obtaining \(x = \frac{5}{4}\).

Step 5: Identify All Solutions

The solutions derived are \(x = 0\) from Step 3, and \(x = \frac{5}{4}\) from Step 4. Thus, the equation \(4x^2 = 5x\) has two solutions: \(x = 0\) and \(x = \frac{5}{4}\).

Key Concepts

Factoring Quadratics

When you encounter a quadratic equation like the one in the exercise, your first step is often to factor the equation. This is because factoring transforms it into a product of simpler expressions, making it easier to solve.
To factor a quadratic expression, recognize any common factors that can be taken out. For the example, the term "4x^2" and "5x" both have "x" in common. We can express the equation as:

• Start with the expression: \(4x^2 - 5x = 0\)
• Factor out the common factor "x": \(x(4x - 5) = 0\)

Now the expression is in its factored form. It's crucial to set the equation to zero when factoring because it aligns with the upcoming step using the zero-product property. Factoring is an essential skill for solving quadratics because it simplifies the equation and sets the stage for finding solutions.

Zero-Product Property

The zero-product property is a crucial tool used after you have factored a quadratic equation. It states that if the product of two factors equals zero, then at least one of the factors must be zero. This property is especially helpful because it reduces complex equations to simpler linear equations.
Applying this to our exercise:

• We have: \(x(4x - 5) = 0\)
• According to the zero-product property, either \(x = 0\) or \(4x - 5 = 0\).

So, even if initially you had a quadratic equation, through factoring and zero-product property, you can convert it into two separate linear equations that are easier to solve. This is what makes the zero-product property a powerful ally in finding the solutions of quadratic equations.

Solving Linear Equations

Once the quadratic equation is broken down into linear parts using factoring and the zero-product property, you solve the resulting linear equations individually.
Take our equation, for example:

• We derived two equations: \(x = 0\) and \(4x - 5 = 0\).
• The first equation, \(x = 0\), is already solved; x is clearly zero.
• For the second equation, \(4x - 5 = 0\), isolate \(x\) by adding 5 to both sides yielding \(4x = 5\).
• Finally, divide both sides by 4 to solve for \(x\), giving you \(x = \frac{5}{4}\).

Linear equations are usually straightforward to solve, and they lead you directly to the roots of the original quadratic equation. Identifying these solutions is the ultimate goal of the problem-solving process for quadratics.