If The Expression $x^3 + 2hx - 2$ Is Equal To 6 When $x = -2$, What Is The Value Of $h$?A. 0 B. 2 C. -4 D. 4

Mar 1, 2025 by ADMIN 118 views

**Solving for the Value of h in a Cubic Expression**

Introduction

In algebra, solving for unknown variables is a crucial aspect of problem-solving. One common type of problem involves substituting given values into an equation and solving for the unknown variable. In this article, we will explore how to solve for the value of $h$ in a cubic expression given a specific value of $x$ . We will use the given expression $x^3 + 2hx - 2$ and the value $x = -2$ to find the value of $h$ .

Understanding the Given Expression

The given expression is a cubic expression, which means it is a polynomial of degree three. The general form of a cubic expression is $ax^3 + bx^2 + cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants. In this case, the expression is $x^3 + 2hx - 2$ , where $h$ is the unknown variable we need to solve for.

Substituting the Given Value of x

To solve for the value of $h$ , we need to substitute the given value of $x$ into the expression. We are given that $x = -2$ , so we will substitute this value into the expression:

(-2)^3 + 2h(-2) - 2

Expanding the Expression

Now that we have substituted the value of $x$ , we need to expand the expression:

-8 - 4h - 2

Simplifying the Expression

Next, we need to simplify the expression by combining like terms:

-10 - 4h

Setting the Expression Equal to 6

We are given that the expression is equal to 6 when $x = -2$ , so we will set the expression equal to 6:

-10 - 4h = 6

Solving for h

Now that we have set the expression equal to 6, we can solve for the value of $h$ . To do this, we need to isolate the variable $h$ on one side of the equation. We can do this by adding 10 to both sides of the equation:

-4h = 16

Dividing Both Sides by -4

Next, we need to divide both sides of the equation by -4 to solve for $h$ :

h = -\frac{16}{4}

Simplifying the Fraction

Finally, we can simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 4:

h = -4

Conclusion

In this article, we used the given expression $x^3 + 2hx - 2$ and the value $x = -2$ to solve for the value of $h$ . We substituted the value of $x$ into the expression, expanded and simplified the expression, set the expression equal to 6, and finally solved for the value of $h$ . The value of $h$ is $-4$ .

Answer

The final answer is $\boxed{-4}$ .

Discussion

This problem is a classic example of how to solve for an unknown variable in a cubic expression. The key steps involved in solving this problem are substituting the given value of $x$ into the expression, expanding and simplifying the expression, setting the expression equal to the given value, and finally solving for the value of $h$ . This problem requires a good understanding of algebraic expressions and equations, as well as the ability to solve for unknown variables.

Q: What is the general form of a cubic expression?

A: The general form of a cubic expression is $ax^3 + bx^2 + cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants.

Q: How do I substitute the given value of x into the expression?

A: To substitute the given value of $x$ into the expression, simply replace $x$ with the given value in the expression. For example, if the expression is $x^3 + 2hx - 2$ and the given value of $x$ is $-2$ , the expression becomes $(-2)^3 + 2h(-2) - 2$ .

Q: How do I expand the expression after substituting the given value of x?

A: To expand the expression, simply multiply out the terms. For example, if the expression is $(-2)^3 + 2h(-2) - 2$ , the expanded expression is $-8 - 4h - 2$ .

Q: How do I simplify the expression after expanding it?

A: To simplify the expression, combine like terms. For example, if the expression is $-8 - 4h - 2$ , the simplified expression is $-10 - 4h$ .

Q: How do I set the expression equal to the given value?

A: To set the expression equal to the given value, simply write the expression equal to the given value. For example, if the expression is $-10 - 4h$ and the given value is 6, the equation is $-10 - 4h = 6$ .

Q: How do I solve for h in the equation?

A: To solve for $h$ , isolate the variable $h$ on one side of the equation. This can be done by adding or subtracting terms from both sides of the equation, and then dividing both sides of the equation by the coefficient of $h$ . For example, if the equation is $-10 - 4h = 6$ , adding 10 to both sides of the equation gives $-4h = 16$ , and then dividing both sides of the equation by -4 gives $h = -4$ .

Q: What if the equation has a fraction? How do I solve for h?

A: If the equation has a fraction, you can solve for $h$ by multiplying both sides of the equation by the denominator of the fraction. For example, if the equation is $-\frac{4h}{3} = 12$ , multiplying both sides of the equation by 3 gives $-4h = 36$ , and then dividing both sides of the equation by -4 gives $h = -9$ .

Q: What if the equation has a negative sign in front of the variable h? How do I solve for h?

A: If the equation has a negative sign in front of the variable $h$ , you can solve for $h$ by multiplying both sides of the equation by -1. For example, if the equation is $-4h = 16$ , multiplying both sides of the equation by -1 gives $4h = -16$ , and then dividing both sides of the equation by 4 gives $h = -4$ .

Q: Can I use a calculator to solve for h?

A: Yes, you can use a calculator to solve for $h$ . Simply enter the equation into the calculator and solve for $h$ . However, be careful to enter the equation correctly and to follow the order of operations.

Q: What if I get a different answer than the one in the solution?

A: If you get a different answer than the one in the solution, double-check your work to make sure you made no mistakes. Check your calculations and make sure you followed the correct steps to solve for $h$ . If you are still unsure, try re-solving the problem or asking a teacher or tutor for help.