Solve for q q^3-q^2-30q=0

This is an algebraic problem where you are asked to find the value(s) of the variable $q$ that satisfy the given cubic equation $q^{3} - q^{2} - 30 q = 0$ . Essentially, the question is asking you to solve for $q$ by factoring the cubic polynomial or by using any other appropriate method to find all possible solutions for $q$ in the equation.

$q^{3} - q^{2} - 30 q = 0$

Answer

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Solution:

Step:1

Begin by factoring the given equation.

Step:1.1

Extract the common factor $q$ from the expression $q^{3} - q^{2} - 30 q$ .

Step:1.1.1

Take $q$ out from $q^{3}$ to get $q \cdot q^{2} - q^{2} - 30 q = 0$ .

Step:1.1.2

Remove $q$ from $- q^{2}$ which gives $q \cdot q^{2} + q (- q) - 30 q = 0$ .

Step:1.1.3

Extract $q$ from $- 30 q$ resulting in $q \cdot q^{2} + q (- q) + q (- 30) = 0$ .

Step:1.1.4

Factor $q$ from $q \cdot q^{2} + q (- q)$ to obtain $q (q^{2} - q) + q (- 30) = 0$ .

Step:1.1.5

Finally, factor $q$ from the entire expression to get $q (q^{2} - q - 30) = 0$ .

Step:1.2

Proceed to factor the quadratic expression.

Step:1.2.1

Apply the AC method to factor $q^{2} - q - 30$ .

Step:1.2.1.1

Identify two numbers that multiply to $- 30$ and add to $- 1$ . These numbers are $- 6$ and $5$ .

Step:1.2.1.2

Express the factored form using these numbers as $q ((q - 6) (q + 5)) = 0$ .

Step:1.2.2

Simplify by removing extraneous parentheses to get $q (q - 6) (q + 5) = 0$ .

Step:2

Recognize that if any factor equals $0$ , the whole equation equals $0$ . Thus, we have $q = 0$ , $q - 6 = 0$ , and $q + 5 = 0$ .

Step:3

Solve for $q$ when $q = 0$ .

Step:4

Find $q$ when $q - 6 = 0$ .

Step:4.1

Set $q - 6$ to $0$ .

Step:4.2

Add $6$ to both sides to find $q = 6$ .

Step:5

Determine $q$ when $q + 5 = 0$ .

Step:5.1

Set $q + 5$ to $0$ .

Step:5.2

Subtract $5$ from both sides to get $q = - 5$ .

Step:6

The complete solution set where $q (q - 6) (q + 5) = 0$ holds true is $q = 0, 6, - 5$ .

Knowledge Notes:

The problem involves solving a cubic equation by factoring. The factoring process is a method of rewriting an expression as a product of its factors. The steps taken include:

Common Factor Extraction: This involves identifying and removing a common factor from all terms in the expression. In this case, $q$ is a common factor.
AC Method: This is a technique used to factor quadratic expressions of the form $a x^{2} + b x + c$ . It requires finding two numbers that multiply to $a c$ and add up to $b$ . These two numbers are then used to break down the middle term and factor by grouping.
Zero Product Property: This property states that if a product of factors equals zero, then at least one of the factors must be zero. This principle is used to find the roots of the equation by setting each factor equal to zero and solving for the variable.
Solving Linear Equations: After factoring, the problem is reduced to solving simple linear equations, which can be done by isolating the variable on one side of the equation.

The solution to the cubic equation is the set of values that satisfy the original equation, which are found by setting each factor equal to zero.