Question

solve the equation.
7x^3 + x^2 - 28x - 4 = 0

the solution set is {}.
(simplify your answer. type an integer or a simplified fraction. use a comma to separate answers as needed.)

Explanation:

Step1: Group the terms

Group the given equation \(7x^{3}+x^{2}-28x - 4=0\) as \((7x^{3}+x^{2})-(28x + 4)=0\).

Step2: Factor out common factors from each group

Factor out \(x^{2}\) from the first - group and 4 from the second - group: \(x^{2}(7x + 1)-4(7x + 1)=0\).

Step3: Factor out the common binomial factor

Factor out \((7x + 1)\): \((7x + 1)(x^{2}-4)=0\).

Step4: Factor the difference of squares

Since \(x^{2}-4=(x + 2)(x - 2)\) (using the formula \(a^{2}-b^{2}=(a + b)(a - b)\) with \(a=x\) and \(b = 2\)), the equation becomes \((7x+1)(x + 2)(x - 2)=0\).

Step5: Set each factor equal to zero and solve for \(x\)

If \(7x+1=0\), then \(7x=-1\), and \(x=-\frac{1}{7}\).
If \(x + 2=0\), then \(x=-2\).
If \(x - 2=0\), then \(x=2\).

Answer:

\(-2,-\frac{1}{7},2\)