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Chapter 1: Problem 12

Solve the equation by factoring, if required: $$ 6 x^{2}=-5 x+6 $$

Short Answer

Expert verified
The solution of the quadratic equation \(6x^2 = -5x + 6\) is \(x = \frac{3}{2}\) and \(x = \frac{-2}{3}\).

Step by step solution

01

Rewrite the equation in the form \(ax^2 + bx + c = 0\)

To rewrite the equation in the form \(ax^2 + bx + c = 0\), we add \(5x\) to both sides and subtract 6 from both sides to obtain: \( 6x^2 + 5x - 6 = 0 \)
02

Factor the quadratic expression

Now we need to factor the quadratic expression \(6x^2 + 5x - 6\). We are looking for two binomials, such that their product is equal to the expression. Two possible factors of 6 (the coefficient of the \(x^2\) term) are 2 and 3. We want these factors (2 and 3) to multiply with two other numbers, such that the sum of the cross-products is equal to the middle term 5x. Through trial and error, we find that the two numbers are -2 and 3. Therefore, the factored form of the quadratic expression is: \( (2x - 3)(3x + 2) = 0 \)
03

Solve for x using the Zero-Product Property

Using the Zero-Product Property, we know that if two factors multiply to equal zero, then one of the factors must be zero. Therefore, we set the factors (2x - 3) and (3x + 2) equal to zero, and solve for x in each case: 1) \( 2x - 3 = 0 \) -> Add 3 to both sides: \( 2x = 3 \) -> Divide both sides by 2: \( x = \frac{3}{2} \) 2) \( 3x + 2 = 0 \) -> Subtract 2 from both sides: \( 3x = -2 \) -> Divide both sides by 3: \( x = \frac{-2}{3} \) The solutions for the equation are \(x = \frac{3}{2}\) and \(x = \frac{-2}{3}\).

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Key Concepts

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

Factoring Quadratics
Factoring quadratics is a technique used to solve equations of the form \( ax^2 + bx + c = 0 \). The goal is to rewrite the quadratic as a product of two binomials. This makes it easier to solve the equation by identifying the roots.

To factor a quadratic expression like \( 6x^2 + 5x - 6 \), you look for two numbers that multiply to the product of the coefficient of \( x^2 \) (which is 6) and the constant term c (which is -6). These numbers also need to add up to 5, the coefficient of the middle term, \( x \).

In our example, the numbers -2 and 3 achieve exactly this. Thus, the quadratic is factored into \((2x - 3)(3x + 2) \).
Zero-Product Property
The Zero-Product Property is a fundamental rule in algebra. It states that if the product of two numbers is zero, then at least one of the numbers must be zero.

For instance, if \( (2x - 3)(3x + 2) = 0 \), then either \( 2x - 3 = 0 \) or \( 3x + 2 = 0 \). This property allows us to set each binomial factor equal to zero and solve for \( x \).

This step is crucial because it moves us from a polynomial expression to numerical solutions, which can be used to determine the roots of the original quadratic equation.
Solving by Factoring
Solving a quadratic equation by factoring is a straightforward method once you have factored the equation. After applying the Zero-Product Property, you solve each of the resulting simple equations.

For example, from \( (2x - 3)(3x + 2) = 0 \), solving the first equation, \( 2x - 3 = 0 \), involves adding 3 to both sides and then dividing by 2 to get \( x = \frac{3}{2} \).

Similarly, solving \( 3x + 2 = 0 \) involves subtracting 2 from both sides and dividing by 3 to find that \( x = \frac{-2}{3} \).
  • Step 1: Set each factor equal to zero.
  • Step 2: Solve the simple linear equations.
  • Step 3: Write down the solutions.
These steps help you find the values of \( x \) that satisfy the original quadratic equation.

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Most popular questions from this chapter

Solve the equation. $$ x^{4}-5 x^{2}+6=0 $$

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