Answer:
To solve the quadratic equation by factoring, we need to set the equation equal to zero and then factor the quadratic expression on the left side of the equation.
The given quadratic equation is:
3x^2 + 4x - 42 = 0
To begin factoring, we look for two numbers that multiply to give -42 (the coefficient of the constant term) and add up to give 4 (the coefficient of the linear term).
After considering the factors of 42, we find that the numbers are 7 and -6.
So, we can rewrite the middle term as 7x - 6x.
Now, let's rewrite the equation using this factored form:
3x^2 + 7x - 6x - 42 = 0
Next, we group the terms and factor them separately:
(3x^2 + 7x) - (6x + 42) = 0
Now, we can factor out the common terms from each group:
x(3x + 7) - 6(3x + 7) = 0
Notice that we have a common binomial term of (3x + 7). We can factor it out:
(3x + 7)(x - 6) = 0
Now, we can apply the zero product property, which states that if a product of factors is equal to zero, then at least one of the factors must be equal to zero.
Setting each factor equal to zero, we have:
3x + 7 = 0 or x - 6 = 0
Solving each equation separately, we find:
3x = -7 or x = 6
Dividing both sides of the first equation by 3, we get:
x = -7/3
So, the solutions to the quadratic equation are:
x = -7/3 or x = 6
Therefore, the correct answer is: x = -7/3, 6.
I hope this explanation helps! Let me know if you have any further questions.
Explanation: