Questions: Solve the following quadratic equation by factoring. Separate multiple answers with a comma. 3 x(x-1)=2(2 x+3)

Solution Steps

To solve the quadratic equation \(3x(x-1) = 2(2x+3)\) by factoring, we first need to expand both sides and bring all terms to one side of the equation to set it to zero. Then, we factor the resulting quadratic expression and solve for the values of \(x\) that satisfy the equation.

Start with the given equation: \[ 3x(x-1) = 2(2x+3) \]

Expand both sides: \[ 3x^2 - 3x = 4x + 6 \]

Bring all terms to one side to set the equation to zero: \[ 3x^2 - 3x - 4x - 6 = 0 \]

Combine like terms: \[ 3x^2 - 7x - 6 = 0 \]

Factor the quadratic expression: \[ 3x^2 - 7x - 6 = (3x + 2)(x - 3) = 0 \]

Set each factor equal to zero and solve for \(x\):

1. \(3x + 2 = 0\) \[ 3x = -2 \] \[ x = -\frac{2}{3} \]

2. \(x - 3 = 0\) \[ x = 3 \]

Final Answer