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Math  /  Algebra

Question2. Expand and simplify the following a) 2(3+x)+4x2(3+x)+4 x b) (x−3)(x+2)(x-3)(x+2) c) (4x−3)2(4 x-3)^{2}

Studdy Solution

STEP 1

1. We need to expand and simplify three separate algebraic expressions.
2. We will use the distributive property and algebraic identities to simplify each expression.

STEP 2

1. Expand and simplify expression (a) 2(3+x)+4x2(3+x)+4x.
2. Expand and simplify expression (b) (x−3)(x+2)(x-3)(x+2).
3. Expand and simplify expression (c) (4x−3)2(4x-3)^2.

STEP 3

Use the distributive property to expand 2(3+x)2(3+x):
2×3+2×x=6+2x 2 \times 3 + 2 \times x = 6 + 2x
Add the result to 4x4x:
6+2x+4x 6 + 2x + 4x
Combine like terms:
6+(2x+4x)=6+6x 6 + (2x + 4x) = 6 + 6x
The simplified expression is:
6+6x \boxed{6 + 6x}

STEP 4

Use the distributive property (FOIL method) to expand (x−3)(x+2)(x-3)(x+2):
First: x×x=x2x \times x = x^2
Outer: x×2=2xx \times 2 = 2x
Inner: −3×x=−3x-3 \times x = -3x
Last: −3×2=−6-3 \times 2 = -6
Combine all terms:
x2+2x−3x−6 x^2 + 2x - 3x - 6
Combine like terms:
x2+(2x−3x)−6=x2−x−6 x^2 + (2x - 3x) - 6 = x^2 - x - 6
The simplified expression is:
x2−x−6 \boxed{x^2 - x - 6}

STEP 5

Use the square of a binomial formula (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2 to expand (4x−3)2(4x-3)^2:
Let a=4xa = 4x and b=3b = 3.
Calculate a2a^2:
(4x)2=16x2 (4x)^2 = 16x^2
Calculate −2ab-2ab:
−2×4x×3=−24x -2 \times 4x \times 3 = -24x
Calculate b2b^2:
32=9 3^2 = 9
Combine all terms:
16x2−24x+9 16x^2 - 24x + 9
The simplified expression is:
16x2−24x+9 \boxed{16x^2 - 24x + 9}

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