Solve The Following Polynomial Equation By Factoring Or Using The Quadratic Formula. Identify All Solutions. X 3 − 9 X 2 + 20 X = 0 X^3 - 9x^2 + 20x = 0 X 3 − 9 X 2 + 20 X = 0

Mar 8, 2025 by ADMIN 176 views

Solve the Following Polynomial Equation by Factoring or Using the Quadratic Formula

===========================================================

Introduction

In this article, we will solve the given polynomial equation by factoring or using the quadratic formula. The equation is $x^3 - 9x^2 + 20x = 0$ . We will first try to factor the equation and then use the quadratic formula if necessary.

Factoring the Equation

To factor the equation, we need to find the greatest common factor (GCF) of the terms. In this case, the GCF is $x$ . We can factor out $x$ from each term:

$x(x^2 - 9x + 20) = 0$

Now, we have a quadratic expression inside the parentheses. We can try to factor it further:

$x(x - 5)(x - 4) = 0$

Using the Quadratic Formula

If we cannot factor the quadratic expression, we can use the quadratic formula to find the solutions. The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 1$ , $b = -9$ , and $c = 20$ . Plugging these values into the formula, we get:

$x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(20)}}{2(1)}$

$x = \frac{9 \pm \sqrt{81 - 80}}{2}$

$x = \frac{9 \pm \sqrt{1}}{2}$

$x = \frac{9 \pm 1}{2}$

Solving for x

Now, we have two possible solutions:

$x = \frac{9 + 1}{2}$

$x = \frac{9 - 1}{2}$

$x = 5$

$x = 4$

Conclusion

In this article, we solved the given polynomial equation by factoring and using the quadratic formula. We found that the solutions are $x = 0$ , $x = 4$ , and $x = 5$ . These solutions satisfy the equation $x^3 - 9x^2 + 20x = 0$ .

Final Answer

The final answer is:

$x = 0, 4, 5$

Step-by-Step Solution

Here is the step-by-step solution to the problem:

Factor out $x$ from each term: $x(x^2 - 9x + 20) = 0$
Factor the quadratic expression: $x(x - 5)(x - 4) = 0$
Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Plug in the values: $x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(20)}}{2(1)}$
Simplify the expression: $x = \frac{9 \pm \sqrt{1}}{2}$
Solve for x: $x = \frac{9 + 1}{2}$ and $x = \frac{9 - 1}{2}$
Find the solutions: $x = 5$ and $x = 4$

Frequently Asked Questions

Q: What is the greatest common factor (GCF) of the terms in the equation?

A: The GCF is $x$ .

Q: How do we factor the quadratic expression?

A: We can factor the quadratic expression by finding two numbers whose product is $20$ and whose sum is $-9$ .

Q: What is the quadratic formula?

A: The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Q: How do we use the quadratic formula to find the solutions?

A: We plug in the values of $a$ , $b$ , and $c$ into the formula and simplify the expression.

Q: What are the solutions to the equation?

A: The solutions are $x = 0$ , $x = 4$ , and $x = 5$ .

====================================================================

Introduction

In this article, we will answer some frequently asked questions about solving polynomial equations. We will cover topics such as factoring, the quadratic formula, and finding solutions.

Q&A

Q: What is the greatest common factor (GCF) of the terms in the equation?

A: The greatest common factor (GCF) is the largest expression that divides each term in the equation without leaving a remainder.

In the case of the equation $x^3 - 9x^2 + 20x = 0$ , the GCF is $x$ . This means that we can factor out $x$ from each term.

Q: How do we factor the quadratic expression?

A: To factor the quadratic expression, we need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

For example, in the quadratic expression $x^2 - 9x + 20$ , we need to find two numbers whose product is $20$ and whose sum is $-9$ . These numbers are $-5$ and $-4$ , so we can factor the expression as $(x - 5)(x - 4)$ .

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that gives the solutions to a quadratic equation.

The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation.

Q: How do we use the quadratic formula to find the solutions?

A: To use the quadratic formula, we need to plug in the values of $a$ , $b$ , and $c$ into the formula and simplify the expression.

For example, in the quadratic equation $x^2 - 9x + 20 = 0$ , we have $a = 1$ , $b = -9$ , and $c = 20$ . Plugging these values into the quadratic formula, we get:

$x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(20)}}{2(1)}$

$x = \frac{9 \pm \sqrt{81 - 80}}{2}$

$x = \frac{9 \pm \sqrt{1}}{2}$

$x = \frac{9 + 1}{2}$ and $x = \frac{9 - 1}{2}$

$x = 5$ and $x = 4$

Q: What are the solutions to the equation?

A: The solutions to the equation are the values of $x$ that make the equation true.

In the case of the equation $x^3 - 9x^2 + 20x = 0$ , the solutions are $x = 0$ , $x = 4$ , and $x = 5$ .

Q: How do we check if the solutions are correct?

A: To check if the solutions are correct, we need to plug them back into the original equation and make sure that the equation is true.

For example, we can plug $x = 0$ back into the equation $x^3 - 9x^2 + 20x = 0$ and get:

$(0)^3 - 9(0)^2 + 20(0) = 0$

$0 - 0 + 0 = 0$

$0 = 0$

This shows that $x = 0$ is a solution to the equation.

Q: What are some common mistakes to avoid when solving polynomial equations?

A: Some common mistakes to avoid when solving polynomial equations include forgetting to factor out the GCF, not using the quadratic formula correctly, and not checking if the solutions are correct.

By avoiding these mistakes, we can ensure that we get the correct solutions to the equation.

Conclusion

In this article, we answered some frequently asked questions about solving polynomial equations. We covered topics such as factoring, the quadratic formula, and finding solutions. By following the steps outlined in this article, we can ensure that we get the correct solutions to the equation.

Final Answer

The final answer is:

$x = 0, 4, 5$

Step-by-Step Solution

Here is the step-by-step solution to the problem:

Factor out the GCF from each term: $x(x^2 - 9x + 20) = 0$
Factor the quadratic expression: $(x - 5)(x - 4) = 0$
Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Plug in the values: $x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(20)}}{2(1)}$
Simplify the expression: $x = \frac{9 \pm \sqrt{1}}{2}$
Solve for x: $x = \frac{9 + 1}{2}$ and $x = \frac{9 - 1}{2}$
Find the solutions: $x = 5$ and $x = 4$

Frequently Asked Questions

Q: What is the greatest common factor (GCF) of the terms in the equation?

A: The greatest common factor (GCF) is the largest expression that divides each term in the equation without leaving a remainder.

Q: How do we factor the quadratic expression?

A: To factor the quadratic expression, we need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that gives the solutions to a quadratic equation.

Q: How do we use the quadratic formula to find the solutions?

A: To use the quadratic formula, we need to plug in the values of $a$ , $b$ , and $c$ into the formula and simplify the expression.

Q: What are the solutions to the equation?

A: The solutions to the equation are the values of $x$ that make the equation true.

Q: How do we check if the solutions are correct?

A: To check if the solutions are correct, we need to plug them back into the original equation and make sure that the equation is true.

Q: What are some common mistakes to avoid when solving polynomial equations?

A: Some common mistakes to avoid when solving polynomial equations include forgetting to factor out the GCF, not using the quadratic formula correctly, and not checking if the solutions are correct.