FORMULA FOR A PLUS B THE WHOLE SQAURE

Expand the following :

Problem 1 :

(3m + 5)²

Solution :

= (3m + 5)²

Using the algebraic identity

(a + b)² = a² + 2ab + b²

Here a = 3m and b = 5

(3m + 5)² = (3m)² + 2(3m)(5) + 5²

 = 3²m² + 30m + 25

= 9m² + 30m + 25

Example 2 :

x² + kx + 9 = (x + a)²

In the equation above, k and are positive constants. If the equation is true all values of x, what is the value of k ?

(a)  0   (b) 6    (c)  3  (d) 9

Solution :

x² + kx + 9 = (x + a)²

Expand (x + a)² using algebraic identity, we get

x² + kx + 9 = x² + 2ax + a²

By comparing the corresponding terms, we get

a² = 9 ---(1) and 2a = k ---(2)

a = 3

By applying the value of a in (2), we get

2(3) = k

k = 6

So, the value of k is 6.

Example 3 :

Which of the following is equivalent to (√a + 2√b)² for all positive values of a and b ?

(a) a - 4b   (b)  a + 4b   (c)  a - 2√ab + 4b  (d) a + 4√ab + 4b

Solution :

(√a + 2√b)²

Here a = √a and b = 2√b

Using the algebraic identity, we get

(√a + 2√b)² = (√a)² + 2(√a)(2√b) + (2√b)²

(√a + 2√b)² = a + 4(√a√b) + (2²√b²)

(√a + 2√b)² = a + 4√ab + 4b

Example 4 :

If a + b = 7 and a² + b² = 31, what is the value of ab?

Solution :

Given, a + b = 7 and a² + b² = 31

By applying the given values in the formula

(a + b)² = a² + 2ab + b²

7² = 31 + 2ab

49 = 31 + 2ab

49 - 31 = 2ab

2ab = 18

Divide by 2, we get

ab = 18/2

ab = 9

Example 5 :

The function f is defined by f(x) = (x - 7)² + 9. If (a-2) = 25. What is one possible value of a ?

Solution :

f(x) = (x - 7)² + 9

f(a - 2) = 25

 Here x = a - 2

f(a-2) = ((a - 2) - 7)² + 9

f(a-2) = (a - 9)² + 9

25 = a² - 18a + 81 + 9

25 = a² - 18a + 90

a² - 18a + 90 - 25 = 0

a² - 18a + 65 = 0

(a - 13)(a - 5) = 0

a = 13 and a = 5

Example 6 :

If (x + 3y)² = x² + 9y² + 42, what us the value of x² y² ?

Solution :

(x + 3y)² = x² + 9y² + 42

x² + 2x(3y) + (3y)² = x² + 9y² + 42

x² + 6xy + 9y² = x² + 9y² + 42

By rearranging the terms in the left side.

x² + 9y² + 6xy = x² + 9y² + 42

6xy = 42

xy = 42/6

xy = 7

Taking squares on both sides, we get

(xy)² = 7²

x² y² = 49

Example 7 :

If n < 0 and 4x² + mx + 9 = (2x + n)², what is the value of m + n ?

Solution :

4x² + mx + 9 = (2x + n)²

4x² + mx + 9 = (2x)² + 2(2x)n + n²

4x² + mx + 9 = 4x² + 4xn + n²

By comparing the corresponding terms, we get

4n = m -----(1) and n² = 9 -----(2)

n = 3, -3

Since n < 0, the value of n is -3

By applying the value of n in (1), we get

m = 4(-3)

m = -12

m + n = -12 + (-3) ==>  -15

So, the value of m + n is -15.

Example 8 :

If (x + 2)² = 4, which of the following is the solution for x ?

(a)  -4   (b)  -2   (c)  2   (d)  8

Solution :

(x + 2)² = 4

x² + 2x(2) + 2² = 4

x² + 4x + 4 = 4

Subtracting 4 on both sides.

x² + 4x = 0

x(x + 4) = 0

x = 0 and x = -4

So, the answer is -4.