Exploring Algebraic Identities

Try and find other patterns like this one. For example, you could consider 4 consecutive squares and see if you can find a pattern.

What can you say about a and b if (a + b)^2 < $a^{2}$ + $b^{2}$?

What can you say about a and b if (a + b)^2 > $a^{2}$ + $b^{2}$?

When will (a + b)^2 be equal to $a^{2}$ + $b^{2}$?

Using the identity (a + b)^2 = $a^{2}$ + 2ab + $b^{2}$, expand the following:

What if we replace b by -b in (a + b)^2 = $a^{2}$ + 2ab + $b^{2}$?

Factor completely:

Find the values of the following using the identity (a - b)^2 = $a^{2}$ - 2ab + $b^{2}$.

Find the following squares using one of the above identities. Determine which of these identities will make these calculations easier.

Factor using suitable identities:

Expand the following using the identity (a + b + c)^2 = $a^{2}$ + $b^{2}$ + $c^{2}$ + 2ab + 2bc + 2ca:

Is this an identity? (a + b - c)^2 + (a - b + c)^2 + (a - b - c)^2 = $2a^{2}$ + $2b^{2}$ + $2c^{2}$.

Try to evaluate the following using a suitable identity:

Suppose 7x is split as 2x + 5x; can a similar rectangular arrangement be formed? Consider other possibilities and check.

Figure out the product of x + 2 and x + 3 using algebra tiles.

Lay out algebra tiles for $x^{2}$ + 11x + 30 in such a way that you will see its factors.

We have seen that (x + 3)(x + 4) = $x^{2}$ + 7x + 12. Also (x + 6)(x + 7) = $x^{2}$ + 13x + 42. Generalise the pattern to get an expression for (x + a)(x + b).

Fill in the blanks with the appropriate expressions to make the equation true. (px + a)(qx + b) = ()$x^{2}$ + ()x + ____. Also, verify your answer using the distributive property.

Fill in the blanks to complete the following identities:

Select and use the identity that will help you to find the following products without multiplying directly: