SAT PRACTICE QUESTIONS USING ALGEBRAIC IDENTITIES

Problem 1 :

(3x + 2y)²

In the expression above can be written as ax² + bxy + cy², where a, b and c are constants, where a + b + c ?

Solution

Problem 2 :

x² + kx + 9 = (x + a)²

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

a)  0      b) 3     c)  6       d)  9

Solution

Problem 3 :

If (x + 3) (x - 3) = 91, what is the value of x² ?

Solution

Problem 4 :

If (c + d) = -5 and (c - d) = -12, then what is the value of c² - d²?

Solution

Problem 5 :

If (mx + c) (nx + 3) = 12x² + 5x - 3 for all values of x, where m, n and c are constants, what is the value of m + n ?

a) 7      b)  8     c)  12       d)  13

Solution

Problem 6 :

In the equation above, a, b and c are constants. If the equation is true for all values of x, what is the value of a + b + c?

Solution

Problem 7 :

If (x + y)² - (x - y)² = 60 and x and y are positive integers, which of the following could be the value of x + y ?

a)  6      b)  8       c)  10    d)  12

Solution

Problem 8 :

(a + b)² - (a - b)²

The expression above equivalent to which of following ?

a) 2ab    b)  4ab     c)  4ab + 2b²      d)  2a² + 2b²

Solution

Problem 9 :

(x - c)² = x + 3

If c = 3, what is the solution set of the equation above ?

a)  {1}    b)  {6}    c)  {1, 6}     d)  {-3, 1, 6}

Solution

Problem 10 :

x² - y² = 48

x + y = 12

If (x, y) is the solution to the system of equations above, what is the value of xy? 

a)  28    b)  32    c)  45    d)  64

Solution

Problem 1 :

If y > 0 and y² - 36 = 0, what is the value of y ?

Solution

Problem 2 :

Which expression is equivalent to 16x⁶ - 24x³y³ + 9y⁶ ?

Solution

Problem 3 :

(k^{x^2 + xy}) (k^{y^2 + xy}) = k²⁵

In the equation above k > 1 and x = 3. What is the positive value of y ?

a)  1    b)  2    c)   4    d)  5

Solution

Problem 4 :

In the equation 3(x - 5)² + 7 = ax² + bx + c, a, b and c are constants. If the equation is true for all values of x, what is the value of c ?

Solution

Problem 5 :

If (cy - d) (cy + d) = 25y² - 16

Which of the following could be the value of c in the equation above, where c and d are constants ?

a)  4     b)  5      c)  16     d)  25

Solution

Problem 6 :

If x² + y² = c  and -xy = b, which of the following is equivalent to c + 2b?

a)  (-2x - y)²    b)  (-x - y)²      c)  (x - y)²      d)   (x + y)² 

Solution

Problem 7 :

Which of the following is an equivalent form of 

(2.6a - 3.5)² - (7.3a² - 4.1)

a)  -2.1a² - 2.9      b)  -2.1a² + 11.1       c)  -0.54a² - 18.2a - 8.15

d)  -0.54a² + 18.2a + 16.35

Solution

Problem 8 :

If x^a² / x^b² = x¹⁶, x > 1 and a + b = 2, what is the value of a - b?

a)  8     b)  14        c)  16        d)  18

Solution

Problem 9 :

9a⁴ + 12a² b² + 4b⁴ 

Which of the following is equivalent to the expression shown above ?

a)  (3a² + 2b²)²            b)  (3a + 2b)⁴        c)  (9a² + 4b²)²          d)  (9a + 4b)⁴

Solution

Problem 10 :

(d − 30)(d + 30) − 7 = −7 What is a solution to the given equation?

Solution