Find the Roots (Zeros) (2x^2+5x-12)/(x+4)

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.

$$2x-3=0\text{or}x+4=0$$

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:

1. 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.

2. Zero of a Fraction: A fraction is zero if and only if its numerator is zero (assuming the denominator is not zero).

3. 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 $x$.

4. 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.

5. 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.

6. 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.