Simplify (2x+3)^2+(x+6)(x-5)
The given problem is asking to combine and simplify two polynomial expressions using algebraic rules and operations. The first expression is a binomial, (2x + 3), being squared which requires application of the binomial square expansion rule (a + b)^2 = a^2 + 2ab + b^2. The second expression is a binomial multiplication (x + 6)(x - 5) which needs to be expanded using the distributive or FOIL method (First, Outside, Inside, Last). After expanding both expressions, like terms are to be combined to simplify the entire expression into its lowest terms or simplest form.
$\left(\left(\right. 2 x + 3 \left.\right)\right)^{2} + \left(\right. x + 6 \left.\right) \left(\right. x - 5 \left.\right)$
Answer
Solution:
Simplify the given expression: $(2x+3)^2+(x+6)(x-5)$
Step 1: Break down each term in the expression.
Step 1.1: Express $(2x+3)^2$ as $(2x+3)(2x+3)$.
$(2x+3)(2x+3) + (x+6)(x-5)$
Step 1.2: Use the FOIL method to expand $(2x+3)(2x+3)$.
Step 1.2.1: Distribute each term.
$2x(2x+3) + 3(2x+3) + (x+6)(x-5)$
Step 1.2.2: Continue distribution.
$2x(2x) + 2x\cdot3 + 3(2x+3) + (x+6)(x-5)$
Step 1.2.3: Complete the distribution.
$2x(2x) + 2x\cdot3 + 3(2x) + 3\cdot3 + (x+6)(x-5)$ $2x(2x) + 2x\cdot3 + 3(2x) + 3\cdot3 + (x+6)(x-5)$
Step 1.3: Combine like terms.
Step 1.3.1: Simplify each term individually.
Step 1.3.1.1: Use the commutative property of multiplication.
$2\cdot2x\cdot x + 2x\cdot3 + 3(2x) + 3\cdot3 + (x+6)(x-5)$
Step 1.3.1.2: Multiply $x$ by $x$ by adding exponents.
Step 1.3.1.2.1: Rearrange the terms.
$2\cdot2(x\cdot x) + 2x\cdot3 + 3(2x) + 3\cdot3 + (x+6)(x-5)$
Step 1.3.1.2.2: Perform the multiplication.
$2\cdot2x^2 + 2x\cdot3 + 3(2x) + 3\cdot3 + (x+6)(x-5)$ $2\cdot2x^2 + 2x\cdot3 + 3(2x) + 3\cdot3 + (x+6)(x-5)$
Step 1.3.1.3: Multiply $2$ by $2$.
$4x^2 + 2x\cdot3 + 3(2x) + 3\cdot3 + (x+6)(x-5)$
Step 1.3.1.4: Multiply $3$ by $2$.
$4x^2 + 6x + 3(2x) + 3\cdot3 + (x+6)(x-5)$
Step 1.3.1.5: Multiply $2$ by $3$.
$4x^2 + 6x + 6x + 3\cdot3 + (x+6)(x-5)$
Step 1.3.1.6: Multiply $3$ by $3$.
$4x^2 + 6x + 6x + 9 + (x+6)(x-5)$ $4x^2 + 6x + 6x + 9 + (x+6)(x-5)$
Step 1.3.2: Add $6x$ and $6x$ together.
$4x^2 + 12x + 9 + (x+6)(x-5)$ $4x^2 + 12x + 9 + (x+6)(x-5)$
Step 2: Expand $(x+6)(x-5)$ using the FOIL method.
Step 2.1: Apply the distributive property.
$4x^2 + 12x + 9 + x(x-5) + 6(x-5)$
Step 2.2: Continue distribution.
$4x^2 + 12x + 9 + x\cdot x + x\cdot(-5) + 6(x-5)$
Step 2.3: Complete the distribution.
$4x^2 + 12x + 9 + x\cdot x + x\cdot(-5) + 6x + 6\cdot(-5)$ $4x^2 + 12x + 9 + x\cdot x + x\cdot(-5) + 6x + 6\cdot(-5)$
Step 3: Simplify and combine like terms.
Step 3.1: Simplify each term.
Step 3.1.1: Multiply $x$ by $x$.
$4x^2 + 12x + 9 + x^2 + x\cdot(-5) + 6x + 6\cdot(-5)$
Step 3.1.2: Rearrange the terms.
$4x^2 + 12x + 9 + x^2 - 5\cdot x + 6x + 6\cdot(-5)$
Step 3.1.3: Multiply $6$ by $-5$.
$4x^2 + 12x + 9 + x^2 - 5x + 6x - 30$ $4x^2 + 12x + 9 + x^2 - 5x + 6x - 30$
Step 3.2: Add $-5x$ and $6x$.
$4x^2 + 12x + 9 + x^2 + x - 30$ $4x^2 + 12x + 9 + x^2 + x - 30$ $4x^2 + 12x + 9 + x^2 + x - 30$
Step 4: Add terms to simplify the expression.
Step 4.1: Add $4x^2$ and $x^2$.
$5x^2 + 12x + 9 + x - 30$
Step 4.2: Combine $12x$ and $x$.
$5x^2 + 13x + 9 - 30$
Step 4.3: Subtract $30$ from $9$.
$5x^2 + 13x - 21$ $5x^2 + 13x - 21$
Knowledge Notes:
To solve this problem, we used several algebraic techniques:
FOIL Method: This is a technique for expanding two binomials. The acronym stands for First, Outer, Inner, Last, referring to the terms in each binomial that are multiplied together.
Distributive Property: This property states that $a(b+c) = ab + ac$. It allows us to multiply a single term by each term inside a set of parentheses.
Combining Like Terms: Terms that have the same variable raised to the same power can be combined by adding or subtracting their coefficients.
Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not change the product, i.e., $ab = ba$.
Simplifying Expressions: This involves performing all possible operations, including addition, subtraction, multiplication, and division, to reduce the expression to its simplest form.
In this problem, we applied these techniques systematically to simplify the given algebraic expression.
