Problem 76 To prepare for Section 12.6, rev... [FREE SOLUTION]

Chapter 12: Problem 76

To prepare for Section 12.6, review solving equations. Solve. 76\. \(5 x^{2}-7 x=0\)

Short Answer

Expert verified

x = 0 or x = \frac{7}{5}

Step by step solution

- Factor out the common term

The given equation is 5x^2 - 7x = 0.To factor out the common term, identify the greatest common factor (GCF) of the terms, which is 'x'. Therefore, we rewrite the equation as: x(5x - 7) = 0.

- Set each factor to zero

According to the zero-product property, if the product of two factors is zero, at least one of the factors must be zero. So, set each factor equal to zero and solve for x: x = 0 5x - 7 = 0

- Solve for x

Solve the equation 5x - 7 = 0 for x. Add 7 to both sides to get: 5x = 7 Divide both sides by 5 to obtain: \[ x = \frac{7}{5} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring

Factoring is a key concept in solving quadratic equations. When we factor, we rewrite the equation as a product of simpler expressions. For example, starting with the equation \(5x^2 - 7x = 0\), we look for a common factor. In this case, the greatest common factor (GCF) is 'x'. We pull 'x' out of each term and rewrite the equation as \(x(5x - 7) = 0\). Factoring simplifies the equation, making it easier to solve.

Zero-Product Property

The zero-product property is a fundamental principle used to solve equations in the form of a product. It states that if the product of two factors is zero, then at least one of the factors must be zero. In our equation \(x(5x - 7) = 0\), this property tells us that either \(x = 0\) or \(5x - 7 = 0\) must be true. By setting each factor to zero, we can solve for the variable.

Greatest Common Factor

The greatest common factor (GCF) is the highest number that can divide all terms in an equation without leaving a remainder. For the equation \(5x^2 - 7x = 0\), the GCF is 'x' because 'x' is the largest common factor in both \(5x^2\) and \(-7x\). Factoring out the GCF simplifies the equation, making the next steps in solving much easier. When you factor out the GCF, you reduce the equation to its simplest form, which is key to solving it.

Solving for Variables

After factoring and applying the zero-product property, we need to solve for the variable. From \(x(5x - 7) = 0\), we get two separate equations: \(x = 0\) and \(5x - 7 = 0\). Solving \(x = 0\) is straightforward. For \(5x - 7 = 0\), add 7 to both sides to get \(5x = 7\). Then, divide by 5 to isolate x: \(x = \frac{7}{5}\). These solutions give us the values of 'x' that satisfy the original equation.

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91影视

To prepare for Section 12.6, review solving equations. Solve. 76\. \(5 x^{2}-7 x=0\)

Short Answer

Step by step solution

- Factor out the common term

- Set each factor to zero

- Solve for x

Key Concepts

Factoring

Zero-Product Property

Greatest Common Factor

Solving for Variables

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