Do you want to learn how to simplify and solve the expression 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6? If so, you have come to the right place. In this blog post, we will explain what the expression means, how to simplify it, how to solve it, and what are some applications and examples of it. We will also provide some tips and tricks to help you master this expression and similar ones.
What does the expression 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6 mean?
The expression 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6 is composed of two parts separated by a semicolon. The first part is “2x^2 – 9x^2,” and the second part is “5 – 3x + y + 6.” The semicolon indicates that the two parts are independent of each other, and can be simplified and solved separately. The expression is a combination of algebraic terms, which are expressions that contain variables, constants, and coefficients. Variables are symbols that represent unknown or changing values, such as x and y. Constants are numbers that do not change, such as 2, 9, 5, and 6. Coefficients are numbers that multiply the variables, such as 2 and -9.
How to simplify the expression 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6?
To simplify the expression 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6, we need to combine the like terms in each part. Like terms are terms that have the same variable and the same exponent, such as 2x^2 and -9x^2. To combine like terms, we need to add or subtract their coefficients, and keep the same variable and exponent. For example, 2x^2 – 9x^2 = (-7)x^2. Here are the steps to simplify the expression:
- Simplify the first part: 2x^2 – 9x^2 = (-7)x^2
- Simplify the second part: 5 – 3x + y + 6 = -3x + y + 11
- Write the simplified expression: (-7)x^2; -3x + y + 11
How to solve the expression 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6?
To solve the expression 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6, we need to find the values of the variables x and y that make the expression true. Since the expression has two parts, we need to solve each part separately, and then find the common solutions for both parts. Here are the steps to solve the expression:
- Solve the first part: (-7)x^2 = 0
- Divide both sides by -7: x^2 = 0
- Take the square root of both sides: x = 0 or x = -0
- The solutions for x are 0 and -0
- Solve the second part: -3x + y + 11 = 0
- Isolate y by adding 3x and subtracting 11 from both sides: y = 3x – 11
- The solution for y is 3x – 11
- Find the common solutions for both parts: x = 0 or x = -0, and y = 3x – 11
- Substitute x = 0 into y = 3x – 11: y = 3(0) – 11 = -11
- Substitute x = -0 into y = 3x – 11: y = 3(-0) – 11 = -11
- The common solutions for x and y are (0, -11) and (-0, -11)
What are some applications and examples of the expression 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6?
The expression 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6 can be used to model and solve some real-world problems, such as:
- Finding the area of a rectangle: If the length of a rectangle is 2x^2 – 9x^2 meters, and the width is 5 – 3x + y + 6 meters, what is the area of the rectangle in square meters?
- To find the area of a rectangle, we need to multiply the length and the width: A = (2x^2 – 9x^2)(5 – 3x + y + 6)
- To simplify the expression, we need to use the distributive property and combine the like terms: A = (-7)x^2(5 – 3x + y + 6) = (-35)x^2 + 21x^3 – 7x^2y – 42x^2
- The area of the rectangle is (-35)x^2 + 21x^3 – 7x^2y – 42x^2 square meters
- Finding the break-even point of a business: If the revenue of a business is 2x^2 – 9x^2 dollars, and the cost is 5 – 3x + y + 6 dollars, what is the break-even point of the business, where the revenue equals the cost?
- To find the break-even point, we need to set the revenue equal to the cost: 2x^2 – 9x^2 = 5 – 3x + y + 6
- To simplify the expression, we need to subtract 5 – 3x + y + 6 from both sides: (-7)x^2 + 3x – y – 11 = 0
- To solve the expression, we need to use the quadratic formula or other methods: x = 0 or x = -0, and y = 3x – 11
- The break-even point of the business is (0, -11) or (-0, -11), where the revenue and the cost are both -11 dollars
What are some tips and tricks to help you master the expression 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6 and similar ones?
Here are some tips and tricks to help you master the expression 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6 and similar ones:
- Remember the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS).
- Remember the distributive property: a(b + c) = ab + ac.
- Remember the quadratic formula: x = (-b ± √(b^2 – 4ac))/2a, where ax^2 + bx + c = 0.
- Remember the difference of squares: a^2 – b^2 = (a + b)(a – b).
- Remember the zero product property: If ab = 0, then a = 0 or b = 0.
- Check your work by plugging in the solutions and verifying that they make the expression true.
Conclusion
The expression 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6 is a combination of algebraic terms that can be simplified and solved separately. To simplify the expression, we need to combine the like terms in each part. To solve the expression, we need to find the values of the variables x and y that make the expression true. The expression can be used to model and solve some real-world problems, such as finding the area of a rectangle or the break-even point of a business. The expression can also be mastered by remembering some rules and formulas, such as the order of operations, the distributive property, the quadratic formula, the difference of squares, and the zero product property.