Square of Difference - Examples, Exercises and Solutions

Understanding Square of Difference

Complete explanation with examples

Difference of Squares Formula

The difference of squares formula is another key algebraic shortcut that simplifies expressions involving two squared terms subtracted from each other. It is written as:

(XY)2=X22XY+Y2(X - Y)^2=X^2 - 2XY + Y^2
This formula skips the need for full expansion and directly factors the expression. It works for both numerical and algebraic expressions, making it versatile in solving equations and simplifying terms. That is, when we encounter two numbers with a minus sign between them, that is, the difference and they will be in parentheses and raised as a squared expression, we can use this formula.

For (xy)2(x - y)^2 the full expansion would be:
(xy)2=(xy)(xy)=xx+x(y)yxy(y)=x22xy+y2(x - y)^2=(x-y)(x−y)=x⋅x+x⋅(-y)-y⋅x−y⋅(-y)=x^2-2xy+y^2

Visual breakdown of abbreviated multiplication formulas: (a+b)² = a² + 2ab + b² and (a−b)² = a² − 2ab + b², with color-coded area models representing the expansion of binomials

Example:

(a4)2=(a - 4)^2=
a×a+a×(4)+(4)×a+(4)×(4)=a\times a+a\times (-4)+ (-4)\times a + (-4) \times (-4) =
a2+2(4a)+(4)2=a^2+2(-4a)+ (-4)^2 =
a28a+16a^2-8a+16

Detailed explanation

Practice Square of Difference

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\( (x-2)^2+(x-3)^2= \)

Examples with solutions for Square of Difference

Step-by-step solutions included
Exercise #1

Declares the given expression as a sum

(7b3x)2 (7b-3x)^2

Step-by-Step Solution

To solve for (7b3x)2(7b - 3x)^2 as a sum, we'll follow these steps:

  • Step 1: Identify the given expression and apply the formula for the square of a difference:
    (ab)2=a22ab+b2(a - b)^2 = a^2 - 2ab + b^2 where a=7ba = 7b and b=3xb = 3x.
  • Step 2: Expand each term:
    • a2=(7b)2=49b2a^2 = (7b)^2 = 49b^2
    • 2ab=2×7b×3x=42bx-2ab = -2 \times 7b \times 3x = -42bx
    • b2=(3x)2=9x2b^2 = (3x)^2 = 9x^2
  • Step 3: Combine all terms to form the sum:
    (7b3x)2=49b242bx+9x2 (7b - 3x)^2 = 49b^2 - 42bx + 9x^2 .

Therefore, the solution to the problem is (7b3x)2=49b242bx+9x2(7b - 3x)^2 = 49b^2 - 42bx + 9x^2.

Hence, the correct answer choice is: 49b242bx+9x2 49b^2 - 42bx + 9x^2

Answer:

49b242bx+9x2 49b^2-42bx+9x^2

Video Solution
Exercise #2

(xx2)2= (x-x^2)^2=

Step-by-Step Solution

To solve the expression (xx2)2(x-x^2)^2, we will use the square of a binomial formula (ab)2=a22ab+b2(a-b)^2 = a^2 - 2ab + b^2.

Let's identify aa and bb in our expression:

  • Here, a=xa = x and b=x2b = x^2.

Applying the formula:

(xx2)2=(x)22(x)(x2)+(x2)2(x - x^2)^2 = (x)^2 - 2(x)(x^2) + (x^2)^2

Calculating each part, we get:

  • (x)2=x2(x)^2 = x^2
  • 2(x)(x2)=2x3-2(x)(x^2) = -2x^3
  • (x2)2=x4(x^2)^2 = x^4

Combining these results, the expression simplifies to:

x42x3+x2x^4 - 2x^3 + x^2

Therefore, the expanded form of (xx2)2(x-x^2)^2 is x42x3+x2\boxed{x^4 - 2x^3 + x^2}.

Answer:

x42x3+x2 x^4-2x^3+x^2

Video Solution
Exercise #3

Choose the expression that has the same value as the following:

(x7)2 (x-7)^2

Step-by-Step Solution

To solve the problem, we need to expand the expression (x7)2(x-7)^2 using the formula for the square of a difference.

The formula for the square of a difference is (ab)2=a22ab+b2(a-b)^2 = a^2 - 2ab + b^2.

Let's apply this formula to our expression (x7)2(x-7)^2:

  • Identify a=xa = x and b=7b = 7.
  • Substitute these values into the formula: (x7)2=x22(x)(7)+72(x-7)^2 = x^2 - 2(x)(7) + 7^2.
  • Calculate each term:
    • x2x^2 remains as x2x^2.
    • 2(x)(7)=14x-2(x)(7) = -14x.
    • 72=497^2 = 49.

So, expanding the expression, we get x214x+49x^2 - 14x + 49.

Thus, the expression that has the same value as (x7)2(x-7)^2 is x214x+49x^2 - 14x + 49.

Answer:

x214x+49 x^2-14x+49

Video Solution
Exercise #4

(a4)(a4)=? (a-4)(a-4)=\text{?}

Step-by-Step Solution

To solve the problem, we will expand the expression (a4)(a4)(a-4)(a-4) using the square of a difference formula.

This formula states: (xy)2=x22xy+y2(x-y)^2 = x^2 - 2xy + y^2.
In our case, x=ax = a and y=4y = 4, so we apply the formula:

  • First term: x2=a2x^2 = a^2
  • Second term: 2xy=2a4=8a-2xy = -2 \cdot a \cdot 4 = -8a
  • Third term: y2=42=16y^2 = 4^2 = 16

Putting it all together, the expression becomes:
a28a+16a^2 - 8a + 16.

After matching this result with the given choices, we find it corresponds to choice 4.

Therefore, the solution to the problem is a28a+16\mathbf{a^2 - 8a + 16}.

Answer:

a28a+16 a^2-8a+16

Video Solution
Exercise #5

Rewrite the following expression as an addition and as a multiplication:

(3xy)2 (3x-y)^2

Step-by-Step Solution

To solve this problem, let's start by identifying the parts of the binomial:

  • The expression (3xy)2(3x-y)^2 represents a binomial squared.
  • We recognize it has the form (ab)2(a-b)^2 where a=3xa = 3x and b=yb = y.
  • Using the formula for the square of a difference: (ab)2=a22ab+b2(a-b)^2 = a^2 - 2ab + b^2, we find the expanded form.

Let's apply the formula:

Step 1: Expand (3xy)2(3x-y)^2 using the formula:
(3xy)2=(3x)22(3x)(y)+y2(3x-y)^2 = (3x)^2 - 2(3x)(y) + y^2

Step 2: Calculate each part:
(3x)2=9x2(3x)^2 = 9x^2
2(3x)(y)=6xy-2(3x)(y) = -6xy
y2y^2 stays as y2y^2

Step 3: Combine these results to get the addition form:
9x26xy+y29x^2 - 6xy + y^2

The expression in multiplication form, as provided, is just repeating the factors:
(3xy)(3xy)(3x-y)(3x-y)

Therefore, the expression rewritten as addition is 9x26xy+y29x^2 - 6xy + y^2 and as multiplication (3xy)(3xy)(3x-y)(3x-y).

Answer:

9x26xy+y2 9x^2-6xy+y^2

(3xy)(3xy) (3x-y)(3x-y)

Video Solution

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