The square of the sum — lesson. Mathematics CBSE, Class 7.

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The square of the sum of two expressions:

{(a + b)}^{2} = a^{2} + 2 ab + b^{2}

The square of the sum of two expressions is equal to the square of the first expression, plus twice the product of the first and the second expression, plus the square of the second expression:

\begin{array}{l} {(a + b)}^{2} = (a + b) \cdot (a + b) = \\ = a \cdot a + a \cdot b + b \cdot a + b \cdot b = \\ = a^{2} + ab + ba + b^{2} = \\ = a^{2} + 2 ab + b^{2} \end{array}

Application of the formula

{(a + b)}^{2} = a^{2} + 2 ab + b^{2}

Example:

1) According to the formula:

\begin{array}{l} {(x + 3)}^{2} = \\ = x^{2} + 2 \cdot x \cdot 3 + 3^{2} = \\ = x^{2} + 6 x + 9 \end{array}

Without the formula (multiplying a polynomial by a polynomial):

\begin{array}{l} {(x + 3)}^{2} = \\ = (x + 3) \cdot (x + 3) = \\ = x \cdot x + x \cdot 3 + 3 \cdot x + 3 \cdot 3 = \\ = x^{2} + 3 x + 3 x + 9 = \\ = x^{2} + 6 x + 9 \end{array}

2) According to the formula:

\begin{array}{l} {(4 x + y)}^{2} = \\ = {(4 x)}^{2} + 2 \cdot 4 x \cdot y + y^{2} = \\ = 16 x^{2} + 8 xy + y^{2} \end{array}

Without the formula (multiplying a polynomial by a polynomial):

\begin{array}{l} {(4 x + y)}^{2} = \\ = (4 x + y) \cdot (4 x + y) = \\ = 4 x \cdot 4 x + 4 x \cdot y + y \cdot 4 x + y \cdot y = \\ = 16 x^{2} + 4 xy + 4 xy + y^{2} = \\ = 16 x^{2} + 8 xy + y^{2} \end{array}

3) According to the formula:

\begin{array}{l} {(6 z + 9)}^{2} = \\ = {(6 z)}^{2} + 2 \cdot 6 z \cdot 9 + 9^{2} = \\ = {36 z}^{2} + 108 z + 81 \end{array}

Without the formula (multiplying a polynomial by a polynomial):

\begin{array}{l} {(6 z + 9)}^{2} = (6 z + 9) \cdot (6 z + 9) = \\ = 6 z \cdot 6 z + 6 z \cdot 9 + 9 \cdot 6 z + 9 \cdot 9 = \\ = 36 z^{2} + 54 z + 54 z + 81 = \\ = 36 z^{2} + 108 z + 81 \end{array}

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2. The square of the sum

Theory: