Problem 74 - Algebra for College Students

Question

Solve each of the equations. $$-4 x^{2}+9 x=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = \frac{9}{4}\).

Step 1: Factor the Equation

We start by factoring the equation. The given equation is \(-4x^2 + 9x = 0\).First, factor out the common term \(x\) from each term:\(x(-4x + 9) = 0\).

Step 2: Set Each Factor to Zero

Since the factored equation \(x(-4x + 9) = 0\) is equal to zero, set each factor equal to zero:1. \(x = 0\)2. \(-4x + 9 = 0\).

Step 3: Solve Each Equation

Solve the equations we obtained from setting each factor to zero:1. For \(x = 0\), the solution is directly \(x = 0\).2. For \(-4x + 9 = 0\), solve for \(x\) as follows: \(-4x = -9\) \(x = \frac{9}{4}\).

Step 4: Verify the Solutions

Finally, verify both solutions by substituting them back into the original equation:1. For \(x = 0\): \(-4(0)^2 + 9(0) = 0\) which holds true.2. For \(x = \frac{9}{4}\): Substitute back: \(-4\left(\frac{9}{4}\right)^2 + 9\left(\frac{9}{4}\right) = -\frac{81}{4} + \frac{81}{4} = 0\). Both solutions hold the original equation.

Key Concepts

Factoring Method

The factoring method is a powerful tool used to solve quadratic equations. A quadratic equation is usually in the form of \( ax^2 + bx + c = 0 \). The idea behind using the factoring method is to express this quadratic equation as a product of its factors. This process often relies on identifying the greatest common factor (GCF) first.

For example, in the equation \(-4x^2 + 9x = 0\), the terms share a common factor of \( x \). By factoring \( x \) out, we simplify the equation to \( x(-4x + 9) = 0 \). This transformation forms two factors: \( x \) and \(-4x + 9 \).

The main goal of factoring is to create an expression where each factor is a polynomial. Once we have factored the equation, we can easily solve for the variable, making it a practical approach to solve quadratic equations.

• Identify the common factors in the terms.
• Rewrite the equation as a product of these factors.

Zero Product Property

The Zero Product Property is a fundamental principle used after factoring a quadratic equation. It states that if a product of two or more terms equals zero, then at least one of the terms must be zero.This property is crucial for solving equations like \( x(-4x + 9) = 0 \). Here, the product is zero, so either \( x = 0 \) or \( -4x + 9 = 0 \). This property simplifies solving since it allows us to split a quadratic equation into simpler linear equations.

Using the Zero Product Property entails the following steps:

• Once the equation is factored, set each individual factor equal to zero.
• Solve these simpler equations independently.

It's an efficient way to discover potential solutions for the variable, especially in quadratic equations, where we expect up to two solutions.

Verifying Solutions

Verifying solutions is an essential final step to ensure the correctness of the solutions obtained through factoring and using the Zero Product Property. For each solution, you substitute back into the original equation to check whether the resulting computation holds true.

In the equation \(-4x^2 + 9x = 0\), after applying the factorization and Zero Product Property, we found \( x = 0 \) and \( x = \frac{9}{4} \) as solutions. To verify:

• Substitute \( x = 0 \) back into the original equation: \(-4(0)^2 + 9(0) = 0\). It holds true, as both terms equal zero.
• Substitute \( x = \frac{9}{4} \): \(-4\left(\frac{9}{4}\right)^2 + 9\left(\frac{9}{4}\right) = -\frac{81}{4} + \frac{81}{4} = 0\). This computation confirms the second solution as well.

Verifying ensures that each solution satisfies the original equation, providing confidence in the solutions' validity.