The difference of squares — lesson. Mathematics CBSE, Class 7.

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The difference of squares of two expressions:

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

The product of the sum and difference of two expressions is equal to the difference of the squares of these expressions:

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

Application of the formula

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

Example:

1) According to the formula:

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

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

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

2) According to the formula:

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

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

\begin{array}{l} (4 x - y) (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} - y^{2} \end{array}

3) According to the formula:

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

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

\begin{array}{l} (6 z - 9) (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} - 81 \end{array}

Did you find an error?

4. The difference of squares