Simplify The Expression:${ X^2 + 15x - 54 }$

Mar 8, 2025 by ADMIN 46 views

**Simplify the Expression: $x^2 + 15x - 54$**

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the various techniques involved in simplifying expressions. In this article, we will focus on simplifying the given quadratic expression: $x^2 + 15x - 54$ . We will use various methods to simplify the expression and provide a step-by-step guide on how to do it.

Understanding the Expression

The given expression is a quadratic expression in the form of $ax^2 + bx + c$ , where $a = 1$ , $b = 15$ , and $c = -54$ . To simplify the expression, we need to factorize it, if possible, or use other techniques such as completing the square or using the quadratic formula.

Factoring the Expression

One of the most common methods of simplifying quadratic expressions is by factoring them. Factoring involves expressing the quadratic expression as a product of two binomial expressions. To factorize the given expression, we need to find two numbers whose product is $-54$ and whose sum is $15$ . These numbers are $18$ and $-3$ , as $18 \times -3 = -54$ and $18 + (-3) = 15$ .

Factoring the Expression: Step-by-Step

To factorize the expression, we can use the following steps:

Identify the numbers: Identify the two numbers whose product is $-54$ and whose sum is $15$ . In this case, the numbers are $18$ and $-3$ .
Write the expression as a product: Write the expression as a product of two binomial expressions, where the first binomial is $x + 18$ and the second binomial is $x - 3$ .
Simplify the expression: Simplify the expression by multiplying the two binomials.

Factoring the Expression: Example

Let's use the above steps to factorize the expression:

$x^2 + 15x - 54$

Identify the numbers: The two numbers whose product is $-54$ and whose sum is $15$ are $18$ and $-3$ .
Write the expression as a product: Write the expression as a product of two binomial expressions:

$x^2 + 15x - 54 = (x + 18)(x - 3)$

Simplify the expression: Simplify the expression by multiplying the two binomials:

$(x + 18)(x - 3) = x^2 - 3x + 18x - 54$

Combine like terms:

$x^2 + 15x - 54$

The expression is already simplified.

Conclusion

In this article, we simplified the given quadratic expression $x^2 + 15x - 54$ by factoring it. We identified the two numbers whose product is $-54$ and whose sum is $15$ , and then wrote the expression as a product of two binomial expressions. We simplified the expression by multiplying the two binomials and combining like terms. This is a crucial skill in mathematics, and it's essential to understand the various techniques involved in simplifying expressions.

Additional Tips and Tricks

Here are some additional tips and tricks to help you simplify quadratic expressions:

Use the quadratic formula: The quadratic formula is a powerful tool for simplifying quadratic expressions. It's given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
Complete the square: Completing the square is another technique for simplifying quadratic expressions. It involves rewriting the expression in the form $(x + \frac{b}{2a})^2 + \frac{c - \frac{b^2}{4a}}$ .
Use algebraic identities: Algebraic identities are powerful tools for simplifying quadratic expressions. Some common algebraic identities include $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$ .

Real-World Applications

Simplifying quadratic expressions has numerous real-world applications. Here are a few examples:

Physics: Quadratic expressions are used to model the motion of objects under the influence of gravity. For example, the equation of motion for an object under the influence of gravity is given by $s = ut + \frac{1}{2}gt^2$ , where $s$ is the displacement, $u$ is the initial velocity, $t$ is the time, and $g$ is the acceleration due to gravity.
Engineering: Quadratic expressions are used to model the behavior of electrical circuits. For example, the equation for the voltage across a resistor is given by $V = IR$ , where $V$ is the voltage, $I$ is the current, and $R$ is the resistance.
Economics: Quadratic expressions are used to model the behavior of economic systems. For example, the equation for the demand for a product is given by $Q = a - bp$ , where $Q$ is the quantity demanded, $a$ is the intercept, $b$ is the slope, and $p$ is the price.

Conclusion

In conclusion, simplifying quadratic expressions is a crucial skill in mathematics, and it's essential to understand the various techniques involved in simplifying expressions. We simplified the given quadratic expression $x^2 + 15x - 54$ by factoring it, and we provided a step-by-step guide on how to do it. We also discussed some additional tips and tricks to help you simplify quadratic expressions, and we provided some real-world applications of simplifying quadratic expressions.

Introduction

In our previous article, we simplified the given quadratic expression $x^2 + 15x - 54$ by factoring it. We identified the two numbers whose product is $-54$ and whose sum is $15$ , and then wrote the expression as a product of two binomial expressions. We simplified the expression by multiplying the two binomials and combining like terms. In this article, we will answer some frequently asked questions (FAQs) related to simplifying quadratic expressions.

Q&A

Q: What is the difference between factoring and simplifying a quadratic expression?

A: Factoring and simplifying a quadratic expression are two different techniques used to manipulate the expression. Factoring involves expressing the quadratic expression as a product of two binomial expressions, while simplifying involves rewriting the expression in a more compact form.

Q: How do I determine if a quadratic expression can be factored?

A: To determine if a quadratic expression can be factored, you need to check if the expression can be written as a product of two binomial expressions. You can do this by looking for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What is the quadratic formula, and how is it used?

A: The quadratic formula is a powerful tool for simplifying quadratic expressions. It's given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . The quadratic formula is used to find the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$ .

Q: What is completing the square, and how is it used?

A: Completing the square is another technique for simplifying quadratic expressions. It involves rewriting the expression in the form $(x + \frac{b}{2a})^2 + \frac{c - \frac{b^2}{4a}}$ . Completing the square is used to rewrite a quadratic expression in a more compact form.

Q: What are some common algebraic identities used in simplifying quadratic expressions?

A: Some common algebraic identities used in simplifying quadratic expressions include $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$ . These identities are used to rewrite a quadratic expression in a more compact form.

Q: How do I apply the quadratic formula to a quadratic equation?

A: To apply the quadratic formula to a quadratic equation, you need to identify the values of $a$ , $b$ , and $c$ in the equation. Then, you can plug these values into the quadratic formula to find the solutions to the equation.

Q: What are some real-world applications of simplifying quadratic expressions?

A: Simplifying quadratic expressions has numerous real-world applications. Some examples include modeling the motion of objects under the influence of gravity, modeling the behavior of electrical circuits, and modeling the behavior of economic systems.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to simplifying quadratic expressions. We discussed the difference between factoring and simplifying a quadratic expression, how to determine if a quadratic expression can be factored, and how to apply the quadratic formula to a quadratic equation. We also discussed some common algebraic identities used in simplifying quadratic expressions and some real-world applications of simplifying quadratic expressions.

Additional Tips and Tricks

Here are some additional tips and tricks to help you simplify quadratic expressions:

Use the quadratic formula: The quadratic formula is a powerful tool for simplifying quadratic expressions. It's given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
Complete the square: Completing the square is another technique for simplifying quadratic expressions. It involves rewriting the expression in the form $(x + \frac{b}{2a})^2 + \frac{c - \frac{b^2}{4a}}$ .
Use algebraic identities: Algebraic identities are powerful tools for simplifying quadratic expressions. Some common algebraic identities include $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$ .

Real-World Applications

Simplifying quadratic expressions has numerous real-world applications. Here are a few examples:

Physics: Quadratic expressions are used to model the motion of objects under the influence of gravity. For example, the equation of motion for an object under the influence of gravity is given by $s = ut + \frac{1}{2}gt^2$ , where $s$ is the displacement, $u$ is the initial velocity, $t$ is the time, and $g$ is the acceleration due to gravity.
Engineering: Quadratic expressions are used to model the behavior of electrical circuits. For example, the equation for the voltage across a resistor is given by $V = IR$ , where $V$ is the voltage, $I$ is the current, and $R$ is the resistance.
Economics: Quadratic expressions are used to model the behavior of economic systems. For example, the equation for the demand for a product is given by $Q = a - bp$ , where $Q$ is the quantity demanded, $a$ is the intercept, $b$ is the slope, and $p$ is the price.