QuestionSolve the quadratic by factoring. $3 x^{2}+12 x=7 x+2$

Studdy Solution

STEP 1

1. The equation $3x^2 + 12x = 7x + 2$ is a quadratic equation.
2. Solving by factoring involves rearranging the equation to set it equal to zero.
3. The quadratic can be factored into two binomials.

STEP 2

1. Rearrange the equation to set it to zero.
2. Factor the quadratic expression.
3. Solve for $x$ by setting each factor equal to zero.

STEP 3

Rearrange the equation to bring all terms to one side, setting the equation to zero. Subtract $7x$ and $2$ from both sides:
$3x^2 + 12x - 7x - 2 = 0$
Simplify the expression:
$3x^2 + 5x - 2 = 0$

STEP 4

Factor the quadratic expression $3x^2 + 5x - 2$ . Look for two numbers that multiply to the product of the leading coefficient and the constant term ( $3 \times -2 = -6$ ) and add to the middle coefficient ( $5$ ).
The numbers $6$ and $-1$ work because $6 \times (-1) = -6$ and $6 + (-1) = 5$ .
Rewrite the middle term using these numbers:
$3x^2 + 6x - x - 2 = 0$
Factor by grouping:
$(3x^2 + 6x) + (-x - 2) = 0$
Factor out the greatest common factor from each group:
$3x(x + 2) - 1(x + 2) = 0$
Factor out the common binomial factor:
$(3x - 1)(x + 2) = 0$

STEP 5

Solve for $x$ by setting each factor equal to zero:
1. $3x - 1 = 0$
2. $x + 2 = 0$
Solve each equation:
1. $3x - 1 = 0$
Add $1$ to both sides:
$3x = 1$
Divide by $3$ :
$x = \frac{1}{3}$
2. $x + 2 = 0$
Subtract $2$ from both sides:
$x = -2$
The solutions to the quadratic equation are:
$x = \frac{1}{3} \quad \text{and} \quad x = -2$

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