Question

98r² - 252rt + 162t²
b) (x + h)² - (x - h)²
= (x + h)(x + h) - (x - h)
= x² + 2xh + h² - x² + 2xh
4xh
c) (10x + 5y)(10x + 5y)
d) (6x - 7y) - (6x + 7y)
e) (x + y)² + (x - y)²

Explanation:

Let's solve each part one by one:

Part (b)

We need to simplify \((x + h)^2 - (x - h)^2\)

Step 1: Expand \((x + h)^2\) and \((x - h)^2\)

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\) and \((a - b)^2 = a^2 - 2ab + b^2\)

\((x + h)^2 = x^2 + 2xh + h^2\)

\((x - h)^2 = x^2 - 2xh + h^2\)

Step 2: Subtract the two expanded forms

\((x + h)^2 - (x - h)^2=(x^2 + 2xh + h^2)-(x^2 - 2xh + h^2)\)

Step 3: Simplify the expression

Remove the parentheses: \(x^2 + 2xh + h^2 - x^2 + 2xh - h^2\)

Combine like terms: \(x^2 - x^2+2xh + 2xh+h^2 - h^2 = 4xh\)

Part (c)

We need to simplify \((10x + 5y)(10x + 5y)\)

Step 1: Recognize the form

This is of the form \((a + b)^2\) where \(a = 10x\) and \(b = 5y\)

Step 2: Apply the formula \((a + b)^2=a^2 + 2ab + b^2\)

\(a^2=(10x)^2 = 100x^2\)

\(2ab = 2\times(10x)\times(5y)=100xy\)

\(b^2=(5y)^2 = 25y^2\)

So \((10x + 5y)^2=100x^2 + 100xy + 25y^2\)

Part (d)

We need to simplify \((6x - 7y)-(6x + 7y)\)

Step 1: Remove the parentheses

\(6x - 7y - 6x - 7y\)

Step 2: Combine like terms

\(6x - 6x-7y - 7y=-14y\)

Part (e)

We need to simplify \((x + y)^2+(x - y)^2\)

Step 1: Expand \((x + y)^2\) and \((x - y)^2\)

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\) and \((a - b)^2 = a^2 - 2ab + b^2\)

\((x + y)^2 = x^2 + 2xy + y^2\)

\((x - y)^2 = x^2 - 2xy + y^2\)

Step 2: Add the two expanded forms

\((x + y)^2+(x - y)^2=(x^2 + 2xy + y^2)+(x^2 - 2xy + y^2)\)

Step 3: Simplify the expression

Combine like terms: \(x^2+x^2 + 2xy- 2xy+y^2 + y^2=2x^2 + 2y^2\)

Part (a) (assuming the first expression is \(98r^2-252rt + 162t^2\))

We need to simplify \(98r^2-252rt + 162t^2\)

Step 1: Factor out the common factor

The greatest common factor of \(98\), \(252\) and \(162\) is \(2\)

\(98r^2-252rt + 162t^2=2(49r^2-126rt + 81t^2)\)

Step 2: Recognize the quadratic form

The expression inside the parentheses \(49r^2-126rt + 81t^2\) is of the form \(a^2-2ab + b^2\) where \(a = 7r\) and \(b = 9t\) (since \((7r)^2=49r^2\), \((9t)^2 = 81t^2\) and \(2\times7r\times9t = 126rt\))

Step 3: Apply the formula \((a - b)^2=a^2 - 2ab + b^2\)

\(49r^2-126rt + 81t^2=(7r - 9t)^2\)

So \(98r^2-252rt + 162t^2=2(7r - 9t)^2\)

Final Answers:

a) \(\boldsymbol{2(7r - 9t)^2}\)

b) \(\boldsymbol{4xh}\)

c) \(\boldsymbol{100x^2 + 100xy + 25y^2}\)

d) \(\boldsymbol{-14y}\)

e) \(\boldsymbol{2x^2 + 2y^2}\)

Answer:

Let's solve each part one by one:

Part (b)

We need to simplify \((x + h)^2 - (x - h)^2\)

Step 1: Expand \((x + h)^2\) and \((x - h)^2\)

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\) and \((a - b)^2 = a^2 - 2ab + b^2\)

\((x + h)^2 = x^2 + 2xh + h^2\)

\((x - h)^2 = x^2 - 2xh + h^2\)

Step 2: Subtract the two expanded forms

\((x + h)^2 - (x - h)^2=(x^2 + 2xh + h^2)-(x^2 - 2xh + h^2)\)

Step 3: Simplify the expression

Remove the parentheses: \(x^2 + 2xh + h^2 - x^2 + 2xh - h^2\)

Combine like terms: \(x^2 - x^2+2xh + 2xh+h^2 - h^2 = 4xh\)

Part (c)

We need to simplify \((10x + 5y)(10x + 5y)\)

Step 1: Recognize the form

This is of the form \((a + b)^2\) where \(a = 10x\) and \(b = 5y\)

Step 2: Apply the formula \((a + b)^2=a^2 + 2ab + b^2\)

\(a^2=(10x)^2 = 100x^2\)

\(2ab = 2\times(10x)\times(5y)=100xy\)

\(b^2=(5y)^2 = 25y^2\)

So \((10x + 5y)^2=100x^2 + 100xy + 25y^2\)

Part (d)

We need to simplify \((6x - 7y)-(6x + 7y)\)

Step 1: Remove the parentheses

\(6x - 7y - 6x - 7y\)

Step 2: Combine like terms

\(6x - 6x-7y - 7y=-14y\)

Part (e)

We need to simplify \((x + y)^2+(x - y)^2\)

Step 1: Expand \((x + y)^2\) and \((x - y)^2\)

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\) and \((a - b)^2 = a^2 - 2ab + b^2\)

\((x + y)^2 = x^2 + 2xy + y^2\)

\((x - y)^2 = x^2 - 2xy + y^2\)

Step 2: Add the two expanded forms

\((x + y)^2+(x - y)^2=(x^2 + 2xy + y^2)+(x^2 - 2xy + y^2)\)

Step 3: Simplify the expression

Combine like terms: \(x^2+x^2 + 2xy- 2xy+y^2 + y^2=2x^2 + 2y^2\)

Part (a) (assuming the first expression is \(98r^2-252rt + 162t^2\))

We need to simplify \(98r^2-252rt + 162t^2\)

Step 1: Factor out the common factor

The greatest common factor of \(98\), \(252\) and \(162\) is \(2\)

\(98r^2-252rt + 162t^2=2(49r^2-126rt + 81t^2)\)

Step 2: Recognize the quadratic form

The expression inside the parentheses \(49r^2-126rt + 81t^2\) is of the form \(a^2-2ab + b^2\) where \(a = 7r\) and \(b = 9t\) (since \((7r)^2=49r^2\), \((9t)^2 = 81t^2\) and \(2\times7r\times9t = 126rt\))

Step 3: Apply the formula \((a - b)^2=a^2 - 2ab + b^2\)

\(49r^2-126rt + 81t^2=(7r - 9t)^2\)

So \(98r^2-252rt + 162t^2=2(7r - 9t)^2\)

Final Answers:

a) \(\boldsymbol{2(7r - 9t)^2}\)

b) \(\boldsymbol{4xh}\)

c) \(\boldsymbol{100x^2 + 100xy + 25y^2}\)

d) \(\boldsymbol{-14y}\)

e) \(\boldsymbol{2x^2 + 2y^2}\)