Solve each equation using the quadratic formula. See Examples 5 and 6.
$(3x+2)(x-1)=3x$

Verified step by step guidance

1. First, expand the left side of the equation by using the distributive property: multiply each term in the first binomial by each term in the second binomial. This gives you \( (3x + 2)(x - 1) = 3x \cdot x + 3x \cdot (-1) + 2 \cdot x + 2 \cdot (-1) \).
2. Simplify the expression from the expansion to get a quadratic expression on the left side: \( 3x^2 - 3x + 2x - 2 \).
3. Combine like terms on the left side to simplify further: \( 3x^2 - x - 2 \).
4. Rewrite the original equation by setting it equal to zero: \( 3x^2 - x - 2 = 3x \). Then subtract \( 3x \) from both sides to get \( 3x^2 - x - 2 - 3x = 0 \), which simplifies to \( 3x^2 - 4x - 2 = 0 \).
5. Identify the coefficients for the quadratic formula: \( a = 3 \), \( b = -4 \), and \( c = -2 \). Then write down the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the values of \( a \), \( b \), and \( c \) into the formula to prepare for solving.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, generally written as ax² + bx + c = 0. Understanding how to rewrite equations into this standard form is essential before applying methods like the quadratic formula.

Expanding and Simplifying Expressions

To solve the given equation, you must first expand the product (3x + 2)(x - 1) and then simplify the resulting expression. This step helps in rearranging the equation into the standard quadratic form.

Quadratic Formula

The quadratic formula x = [-b ± √(b² - 4ac)] / (2a) provides the solutions to any quadratic equation ax² + bx + c = 0. It requires identifying coefficients a, b, and c correctly and calculating the discriminant to find real or complex roots.