Question

Solve each equation. $$ (3 x+2)(x-3)=7 x-1 $$

   Solve each equation.
$$
(3 x+2)(x-3)=7 x-1
$$

Intermediate Algebra

Intermediate Algebra

John Hornsby; Terry… 13th Edition

Chapter 5, Problem 34

Instant Answer

Solved by Expert Mihir Gupta

02/23/2022

Step 1

So, we have: \[ (3x+2)(x-3) = 7x - 1 \] which simplifies to \[ 3x^2 - 7x - 6 = 7x - 1 \] and further simplifies to \[ 3x^2 - 14x - 5 = 0 \] Show more…

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Solve each equation. $$ (3 x+2)(x-3)=7 x-1 $$

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Key Concepts

Expanding Expressions

This concept involves using the distributive property to multiply two binomials, which transforms a factored expression into a standard polynomial form. It is essential for turning the original equation into a form that can be simplified and further manipulated.

Rearranging to Standard Form

After expanding the expression, the next step is to collect like terms and rearrange the equation so that all terms are on one side, setting the equation equal to zero. This standard form (ax² + bx + c = 0) is critical for applying techniques for solving quadratic equations.

Solving Quadratic Equations

Once the equation is in standard form, various methods such as factoring, completing the square, or using the quadratic formula can be applied to determine the values of the variable. This concept is fundamental to solving second-degree polynomial equations.

Factoring

Factoring is a method used to express the quadratic equation as a product of two linear factors. When the quadratic is factorable, this method provides a straightforward way to identify the solutions by setting each factor equal to zero.

Zero Product Property

This property states that if a product of two factors equals zero, then at least one of the factors must be zero. It is a key concept used after factoring to split the quadratic equation into simpler equations that can be solved individually.

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