PRACTICE PROBLEMS ON QUADRATIC EQUATIONS FOR SAT

Problem 1 :

In the xy-plane, what is the distance between the two x-intercepts of the parabola

y = x² - 3x - 10 ?

a)  3      b)  5      c)  7    d)  10

Solution

Problem 2 :

What are the solutions to x² + 4x + 2 = 0 ?

a) -2 ± √2     b) 2 ± 2√2      c) -2 ± 2√2   d) -4 ± 2√2

Solution

Problem 3 :

If a < 1 and 2a² - 7a + 3 = 0, what is the value of a ?

Solution

Problem 4 :

3x² + 10x = 8

If a and b are two solutions to the equation above and a > b, what is the value of b² ?

a)  4/9    b)  2/3    c) 4    d)  16

Solution

Problem 5 :

What is the sum of the solutions of (2x- 3)² = 4x + 5 ?

Solution

Problem 6 :

y = -3

y = x² + cx

In the system of equations, c is a constant. For which of the following values of c does  the system of equations have exactly two real solutions ?

a)  -4    b)  1    c) 2    d)  3

Solution

Problem 7 :

At which of the following points does the line with equation y = 4 intersect the parabola y = (x + 2)² - 5 in the xy-plane ?

a)  (-1, 4) and (-5, 4)      b)  (1, 4) and (-5, 4)

c)  (1, 4) and (5, 4)        d)  (-11, 4) and (7, 4)

Solution

Problem 8 :

Which of the following equations represents the parabola shown in the xy-plane above ?

a)  y = (x - 3)² - 8           b)  y = (x + 3)² + 8

c)  y = 2(x - 3)² - 8         d)  y = 2(x + 3)² - 8

Solution

Problem 9 :

For what value of t does the equation v = 5t - t² result in the maximum value of v ?

Solution

Problem 10 :

P = m² - 100m - 120000

The monthly profit of a mattress company cab be modeled by the equation above, where P is the profit in dollars, and m is the number of mattresses the company must sell in the given month so that it does not lose money during that month ?

Solution

Problem 11 :

y = -3

y = ax² + 4x - 4

In the system if equations above, a is a constant. For which of the following of a does the system of equations have exactly one real solution ?

a)  -4        b)  -2        c)  2       d) 4

Solution

Problem 12 :

f(x) = -x² + 6x + 20

The function f is defined above, which of the following is equivalent form of f(x) displays the maximum value of f as a constant or coefficient ?

a)  f(x) = -(x - 3)² + 11          b)  f(x) = -(x - 3)² + 29  

c)  f(x) = -(x + 3)² + 11        d)  f(x) = -(x + 3)² + 29

Solution

Problem 13 :

y = a(x - 3)(x - k)

In the quadratic equation above a and k are constants. If the graph of the equation in the xy-plane is a parabola with vertex (5, -32). What is the value of a ?

a)  2     b)  5       c)  6      d) 8

Solution

Problem 14 :

In the xy-plane, he line y = 2x + b intersects the parabola y = x² + bx + 5 at the point (3, k). If b is a constant, what is the value of k ?

Solution