Question

First, square each of the binomials and place them into perfect square trinomial form, and then subtract the trinomials.

10. (x + 3)^2 - (x - 4)^2

13. (7b^2 - 7)^2 - (2b^2 - 11)^2

14.(5ab^2 + 3)^2 - (6ab^2 - 7)^2

Answer 1

(a) 14x - 7

(b) 45b⁴ - 54b² - 72

(c) -11a²b⁴ + 114ab² - 40

Explanation:

To simplify the given expressions, we'll first expand and simplify the square of each binomial, and then apply the distributive property to subtract them.

Let's simplify each of the following expression step-by-step:

(a) (x + 3)² - (x - 4)²

(b) (7b² - 7)² - (2b² - 11)²

(c) (5ab² + 3)² - (6ab² - 7)²

`\hrulefill`

Part (a): (x + 3)² - (x - 4)²

Step 1: Square each binomial.

⇒(x + 3)² = (x + 3)(x + 3) = x² + 6x + 9

⇒ (x - 4)² = (x - 4)(x - 4) = x² - 8x + 16

Step 2: Subtract the trinomials.

⇒ (x² + 6x + 9) - (x² - 8x + 16)

Now, apply the distributive property to distribute the negative sign across the second trinomial:

⇒ x² + 6x + 9 - x² + 8x - 16

Step 3: Simplify by combining like terms.

⇒ (x² - x²) + (6x + 8x) + (9 - 16)

⇒ (0) + (14x) + (-7)

∴ 14x - 7

So, the simplified expression is 14x - 7.

We will repeat this step-by-step process for parts (b) and (c).

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Part (b): (7b² - 7)² - (2b² - 11)²

Step 1:

⇒ (7b² - 7)² = (7b² - 7)(7b² - 7) = 49b⁴ - 98b² + 49

⇒ (2b² - 11)² = (2b² - 11)(2b² - 11) = 4b⁴ - 44b² + 121

Step 2:

⇒ (49b⁴ - 98b² + 49) - (4b⁴ - 44b² + 121)

⇒ 49b⁴ - 98b² + 49 - 4b⁴ + 44b² - 121

Step 3:

⇒ (49b⁴ - 4b⁴) + (- 98b² + 44b²) + (49 - 121)

∴ 45b⁴ - 54b² - 72

So, the simplified expression is 45b⁴ - 54b² - 72.

`\hrulefill`

Part (c): (5ab² + 3)² - (6ab² - 7)²

Step 1:

⇒ (5ab² + 3)² = (5ab² + 3)(5ab² + 3) = 25a²b⁴ + 30ab² + 9

⇒ (6ab² - 7)² = (6ab² - 7)(6ab² - 7) = 36a²b⁴ - 84ab² + 49

Step 2:

⇒ (25a²b⁴ + 30ab² + 9) - (36a²b⁴ - 84ab² + 49)

⇒ 25a²b⁴ + 30ab² + 9 - 36a²b⁴ + 84ab² - 49

Step 3:

⇒ (25a²b⁴ - 36a²b⁴) + (30ab² + 84ab²) + (9 - 49)

∴ -11a²b⁴ + 114ab² - 40

So, the simplified expression is -11a²b⁴ + 114ab² - 40.

All parts have been solved.

`\hrulefill`

Additional Information:

Perfect Square Trinomial: A perfect square trinomial is a trinomial of the form (a + b)² or (a - b)². When you square a binomial, you can use the formula (a + b)² = a² + 2ab + b² or (a - b)² = a² - 2ab + b² to expand it.

Combining Like Terms: When simplifying algebraic expressions, combine like terms by adding or subtracting the coefficients of the same variables. In the examples above, we combined the terms with the same variable and exponent to simplify the expressions.

Distributive property: The distributive property is a mathematical rule that states multiplying a number by the sum (or difference) of two other numbers is the same as multiplying the number by each term separately and then adding (or subtracting) the results. In algebraic terms, a · (b + c) = (a · b) + (a · c) and a · (b - c) = (a · b) - (a · c).

Remember, practice makes perfect! Working through more examples will help reinforce these concepts and improve your algebraic skills.