Simplify: 7x + 3x(x – 4) = ? A. 10x + 12 B. 10x – 12 C. 3x2 + 5x
D
Simplifying Algebraic Expressions: Step-by-Step Solution
This problem asks us to simplify the given algebraic expression: \(7x + 3x(x – 4)\).
To simplify this expression, we need to follow the order of operations, which involves first dealing with the multiplication part before combining like terms.
Step 1: Distribute the term outside the parentheses
We have \(3x\) multiplied by the terms inside the parentheses \((x - 4)\). We distribute \(3x\) to both \(x\) and \(-4\):
The multiplication is: \(3x \times (x - 4)\)
This expands to: \((3x \times x) + (3x \times -4)\)
Calculating the products:
- \(3x \times x = 3x^2\)
- \(3x \times -4 = -12x\)
So, \(3x(x – 4)\) simplifies to \(3x^2 - 12x\).
Step 2: Rewrite the original expression with the simplified part
Now substitute the simplified part back into the original expression:
The original expression is: \(7x + 3x(x – 4)\)
Substitute \(3x(x – 4)\) with \(3x^2 - 12x\):
Expression becomes: \(7x + 3x^2 - 12x\)
Step 3: Combine like terms
In the expression \(7x + 3x^2 - 12x\), we need to identify terms that have the same variable raised to the same power. The terms \(7x\) and \(-12x\) are like terms because they both contain the variable \(x\) raised to the power of 1. The term \(3x^2\) is not a like term with these because it has \(x\) raised to the power of 2.
Combine the like terms \(7x\) and \(-12x\):
\(7x - 12x = (7 - 12)x = -5x\)
Step 4: Write the final simplified expression
Now, put all the terms together. It's standard practice to write polynomials in descending order of powers of the variable.
We have the terms \(3x^2\) and \(-5x\).
The simplified expression is: \(3x^2 - 5x\)
Checking the Options
Let's compare our simplified expression \(3x^2 - 5x\) with the given options:
- A. \(10x + 12\)
- B. \(10x – 12\)
- C. \(3x^2 + 5x\)
- D. \(3x^2 – 5x\)
Our result \(3x^2 - 5x\) matches option D.
| Step | Action | Expression |
|---|---|---|
| Start | Original expression | \(7x + 3x(x - 4)\) |
| 1 | Distribute \(3x\) | \(7x + (3x \times x) + (3x \times -4)\) |
| Simplify multiplication | \(7x + 3x^2 - 12x\) | |
| 2 | Identify like terms | \(\underline{7x} + 3x^2 \underline{- 12x}\) |
| 3 | Combine like terms | \(3x^2 + (7x - 12x)\) |
| Simplify | \(3x^2 - 5x\) | |
| Final | Simplified form | \(3x^2 - 5x\) |
Revision Table: Key Concepts for Simplifying Expressions
| Concept | Description | Example |
|---|---|---|
| Distributive Property | Multiply a term outside parentheses by each term inside. \(a(b+c) = ab + ac\) | \(2(x+3) = 2x + 6\) |
| Like Terms | Terms with the same variable(s) raised to the same power(s). | \(5x\) and \(-2x\) are like terms; \(3x^2\) and \(4x\) are not. |
| Combining Like Terms | Add or subtract the coefficients of like terms while keeping the variable part the same. | \(5x - 2x = (5-2)x = 3x\) |
Additional Information: Polynomials and Terms
The expression we simplified is a polynomial. A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
- Term: A single number, variable, or the product of a number and one or more variables raised to non-negative integer powers (e.g., \(7x\), \(3x^2\), \(-12x\)).
- Coefficient: The numerical factor of a term (e.g., in \(7x\), the coefficient is 7; in \(3x^2\), the coefficient is 3; in \(-12x\), the coefficient is -12).
Simplifying algebraic expressions involves combining like terms and applying properties like the distributive property to write the expression in its most compact form.
Important Questions from Linear Equation in 2 or more Variables
If \(3x+6y+9z = \dfrac{20}{3}, 6x+9y + 3z = \dfrac{17}{3}\) and \(18x+ 27y - z = \dfrac{113}{9}\) , then what is the value of \(75x+113y \ ?\)
If the system of equations 2x - 3y - 3 and -4x + qy - p/2 is inconsistent which of the following cannot be the value of p ?
If \(4x + \dfrac{6}{y}= 15 \) and \(6x - \dfrac{8}{y} = 14\) , then the value of p in y = px - 2 is :
If 5x + y = 17 and xy = 6, then what is the value of 125x3 + y3 ?
Two mixers and one TV cost Rs. 500, while two TVs and one mixer cost Rs. 700. The cost of one TV is:
