Simplifying Another Expression

Oct 13, 2025 by pynkocean.com 6 views

Introduction

Equations and expressions are fundamental concepts in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific equation and simplifying expressions without using a calculator. We will break down the problem into manageable steps and provide a clear explanation of each step.

Solving the Equation

The given equation is:

${ -\sqrt{5-x} = x + 1 }$

To solve this equation, we need to isolate the variable x. We can start by squaring both sides of the equation to eliminate the square root.

Squaring Both Sides

When we square both sides of the equation, we get:

${ (-\sqrt{5-x})^2 = (x + 1)^2 }$

Using the property of exponents, we can simplify the left-hand side of the equation:

${ -(5-x) = x^2 + 2x + 1 }$

Now, we can simplify the equation further by distributing the negative sign on the left-hand side:

${ -5 + x = x^2 + 2x + 1 }$

Rearranging the Equation

To isolate the variable x, we need to move all the terms to one side of the equation. We can do this by subtracting x^2, 2x, and 1 from both sides of the equation:

${ -5 + x - x^2 - 2x - 1 = 0 }$

Simplifying the equation further, we get:

${ -x^2 - x - 6 = 0 }$

Solving the Quadratic Equation

The equation we obtained is a quadratic equation in the form of ax^2 + bx + c = 0. We can solve this equation using the quadratic formula:

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

In this case, a = -1, b = -1, and c = -6. Plugging these values into the quadratic formula, we get:

${ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-1)(-6)}}{2(-1)} }$

Simplifying the equation further, we get:

${ x = \frac{1 \pm \sqrt{1 - 24}}{-2} }$

${ x = \frac{1 \pm \sqrt{-23}}{-2} }$

Since the square root of a negative number is not a real number, we can conclude that this equation has no real solutions.

Simplifying Expressions

Now, let's move on to simplifying expressions. We will start with the first expression:

${ \left(2 \frac{1}{4}\right)^{-\frac{1}{2}} }$

Simplifying the Expression

To simplify this expression, we need to follow the order of operations (PEMDAS):

Evaluate the expression inside the parentheses: ${ 2 \frac{1}{4} = \frac{9}{4} }$
Raise the result to the power of -1/2: ${ \left(\frac{9}{4}\right)^{-\frac{1}{2}} = \left(\frac{4}{9}\right)^{\frac{1}{2}} }$
Simplify the expression: ${ \left(\frac{4}{9}\right)^{\frac{1}{2}} = \frac{2}{3} }$

Therefore, the simplified expression is:

${ \frac{2}{3} }$

Simplifying Another Expression

Now, let's simplify the second expression:

${ \frac{6^{2-x} \times 3^{x-1}}{2^{x-1}} }$

Simplifying the Expression

To simplify this expression, we need to follow the order of operations (PEMDAS):

Evaluate the expressions inside the parentheses: ${ 6^{2-x} = 6^{2} \times 6^{-x} = 36 \times 6^{-x} }$ ${ 3^{x-1} = 3^{x} \times 3^{-1} = 3^{x} \times \frac{1}{3} }$ ${ 2^{x-1} = 2^{x} \times 2^{-1} = 2^{x} \times \frac{1}{2} }$
Substitute the results back into the expression: ${ \frac{36 \times 6^{-x} \times 3^{x} \times \frac{1}{3}}{2^{x} \times \frac{1}{2}} }$
Simplify the expression: ${ \frac{36 \times 6^{-x} \times 3^{x} \times \frac{1}{3}}{2^{x} \times \frac{1}{2}} = \frac{36 \times 6^{-x} \times 3^{x}}{2^{x}} }$
Simplify further: ${ \frac{36 \times 6^{-x} \times 3^{x}}{2^{x}} = \frac{36 \times 3^{x}}{2^{x} \times 6^{x}} }$
Simplify further: ${ \frac{36 \times 3^{x}}{2^{x} \times 6^{x}} = \frac{6^{2} \times 3^{x}}{2^{x} \times 6^{x}} }$
Simplify further: ${ \frac{6^{2} \times 3^{x}}{2^{x} \times 6^{x}} = \frac{6^{2-x}}{2^{x}} }$
Simplify further: ${ \frac{6^{2-x}}{2^{x}} = 6^{2-x} \times 2^{-x} }$
Simplify further: ${ 6^{2-x} \times 2^{-x} = \frac{6^{2-x}}{2^{x}} }$

Therefore, the simplified expression is:

${ \frac{6^{2-x}}{2^{x}} }$

Conclusion

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ax^2 + bx + c = 0. The formula is:

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of a, b, and c in the quadratic equation. Then, plug these values into the formula and simplify.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, which means it has a squared variable (x^2). A linear equation, on the other hand, is a polynomial equation of degree one, which means it has a variable (x) but no squared variable.

Q: How do I simplify an expression?

A: To simplify an expression, you need to follow the order of operations (PEMDAS):

Evaluate the expressions inside the parentheses.
Exponentiate (raise to a power).
Multiply and divide from left to right.
Add and subtract from left to right.

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is a fraction that contains only rational numbers (numbers that can be expressed as a ratio of integers). An irrational expression, on the other hand, is a fraction that contains at least one irrational number (a number that cannot be expressed as a ratio of integers).

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to follow these steps:

Factor the numerator and denominator.
Cancel out any common factors.
Simplify the resulting fraction.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that contains a linear expression (an expression with a variable but no squared variable). A quadratic inequality, on the other hand, is an inequality that contains a quadratic expression (an expression with a squared variable).

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign. You can do this by adding or subtracting the same value to both sides of the inequality.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the expression to determine the solution set.

Q: What is the difference between a system of linear equations and a system of quadratic equations?

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously. A system of quadratic equations, on the other hand, is a set of two or more quadratic equations that are solved simultaneously.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to use the method of substitution or elimination to find the values of the variables that satisfy all the equations.

Q: How do I solve a system of quadratic equations?

A: To solve a system of quadratic equations, you need to use the method of substitution or elimination to find the values of the variables that satisfy all the equations. You can also use the quadratic formula to solve the equations.

Conclusion

In this article, we have answered some frequently asked questions about solving equations and simplifying expressions. We hope this article has provided a clear explanation of the concepts and has helped you to better understand how to solve equations and simplify expressions.