Find the Roots (Zeros) (2x^2+5x-12)/(x+4)
The problem asks for the calculation of the values of the variable x for which the given rational function (2x^2+5x-12)/(x+4) becomes zero. In other words, you need to determine the x-values that turn the numerator of the function, which is a quadratic polynomial (2x^2+5x-12), into zero, since the denominator being (x+4) must not be equal to zero for the function to remain defined. Essentially, you are being tasked with finding the zeros or roots of the quadratic equation 2x^2 + 5x - 12 = 0 while ensuring that the solutions do not include x = -4, as that would make the denominator zero, leading to an undefined expression.
$\frac{2 x^{2} + 5 x - 12}{x + 4}$
Answer
Solution:
Step 1: Equate the Rational Expression to Zero
Begin by setting the rational expression $\frac{2x^2 + 5x - 12}{x + 4}$ equal to zero.
$$\frac{2x^2 + 5x - 12}{x + 4} = 0$$
Step 2: Find the Values of x
Step 2.1: Focus on the Numerator
Since a fraction equals zero when its numerator is zero, set the numerator to zero.
$$2x^2 + 5x - 12 = 0$$
Step 2.2: Factor the Quadratic Equation
Step 2.2.1: Use Factor by Grouping
Step 2.2.1.1: Split the Middle Term
Split the middle term into two terms that multiply to $2 \cdot -12 = -24$ and add up to $5$.
Step 2.2.1.1.1: Extract the Factor from the Middle Term
Isolate the $5x$ term.
$$2x^2 + 5(x) - 12 = 0$$
Step 2.2.1.1.2: Decompose the Middle Term
Decompose $5x$ into $-3x + 8x$.
$$2x^2 + (-3x + 8x) - 12 = 0$$
Step 2.2.1.1.3: Distribute and Combine Like Terms
Apply distribution to combine like terms.
$$2x^2 - 3x + 8x - 12 = 0$$
Step 2.2.1.2: Group and Factor
Step 2.2.1.2.1: Create Two Bins
Group the terms into two pairs.
$$(2x^2 - 3x) + (8x - 12) = 0$$
Step 2.2.1.2.2: Factor Out the Common Factors
Factor out the common factor from each bin.
$$x(2x - 3) + 4(2x - 3) = 0$$
Step 2.2.1.3: Factor Out the Common Binomial
Factor out the common binomial factor $(2x - 3)$.
$$(2x - 3)(x + 4) = 0$$
Step 2.2.2: Apply the Zero Product Property
For the product to be zero, one of the factors must be zero.
$$2x - 3 = 0 \quad \text{or} \quad x + 4 = 0$$
Step 2.2.3: Solve the First Factor
Step 2.2.3.1: Isolate the x-term
Set the first factor equal to zero.
$$2x - 3 = 0$$
Step 2.2.3.2: Solve for x
Add $3$ to both sides and then divide by $2$.
$$x = \frac{3}{2}$$
Step 2.2.4: Solve the Second Factor
Step 2.2.4.1: Isolate the x-term
Set the second factor equal to zero.
$$x + 4 = 0$$
Step 2.2.4.2: Solve for x
Subtract $4$ from both sides.
$$x = -4$$
Step 2.2.5: Combine the Solutions
The solutions are the values of $x$ that satisfy the equation.
$$x = \frac{3}{2}, -4$$
Step 2.3: Exclude Non-valid Solutions
Exclude any solutions that do not satisfy the original rational expression.
$$x = \frac{3}{2}$$
Step 3: Conclusion
The root of the rational expression is $x = \frac{3}{2}$.
Knowledge Notes:
To find the roots (also known as zeros) of a rational expression, you must set the expression equal to zero and solve for the variable. Here are the key steps and concepts used in the problem-solving process:
Rational Expressions: A rational expression is a fraction where the numerator and the denominator are polynomials. The roots of the rational expression are the values of the variable that make the expression equal to zero.
Zero of a Fraction: A fraction is zero if and only if its numerator is zero (assuming the denominator is not zero).
Factoring Quadratics: To solve a quadratic equation, you can factor it into the product of two binomials if possible. This often involves finding two numbers that multiply to give the product of the coefficient of $x^2$ and the constant term, and add up to the coefficient of $x$.
Factor by Grouping: This is a method used to factor certain types of polynomials. It involves rearranging the terms and factoring out the greatest common factor from each group.
Zero Product Property: If the product of two expressions is zero, then at least one of the expressions must be zero. This property allows us to set each factor of a factored equation equal to zero to find the roots.
Excluding Extraneous Solutions: In the context of rational expressions, some solutions may not be valid because they make the denominator zero. These solutions must be excluded from the final answer.
