In the xy-coordinate plane, the graph of the equation y = 3x² - 12x - 36 has zeros at x = a and x = b,

where a < b. The graph has a minimum at (c , -48). What are the values of a, b, and c?

A.) a = 2, b = 4, and c = 2

B.) a = -2, b = 6, and c = 2

C.) a = -3, b = 3, and c = 0

D.) a = 3, b = 6, and c = 2

where a < b. The graph has a minimum at (c , -48). What are the values of a, b, and c?

A.) a = 2, b = 4, and c = 2

B.) a = -2, b = 6, and c = 2

C.) a = -3, b = 3, and c = 0

D.) a = 3, b = 6, and c = 2

y = 3x² - 12x - 36

3x² - 12x - 36 = 0 {substituted 0 for y}

3(x² - 4x - 12) = 0 {factored out the greatest common factor, 3}

x² - 4x - 12 = 0 {divided each side by 3}

(x - 6)(x + 2) = 0 {factored into two binomials}

x - 6 = 0 or x + 2 = 0 {set each factor equal to 0}

x = 6 or x = -2 {added 6 and subtracted 2, respectively}

Since a < b

a = -2 and b = 6

The minimum, is the y-coordinate of the vertex, -48. To find the the value of the x-coordinate of the

vertex, c, substitute -48 in for y into the equation y = 3x² - 12x - 36 and solve for x.

-48 = 3x² - 12x - 36

3x² - 12x + 12 = 0 {added 48 to each side}

3(x² - 4x + 4) = 0 {factored out the greatest common factor, 3}

x² - 4x + 4 = 0 {divided each side by 3}

(x - 2)(x - 2) = 0 {factored into two binomials}

x - 2 = 0 {set the factor equal to 0}

x = 2 {added 2 to each side}

Therefore, the x-coordinate of the vertex, c, is 2.

3x² - 12x - 36 = 0 {substituted 0 for y}

3(x² - 4x - 12) = 0 {factored out the greatest common factor, 3}

x² - 4x - 12 = 0 {divided each side by 3}

(x - 6)(x + 2) = 0 {factored into two binomials}

x - 6 = 0 or x + 2 = 0 {set each factor equal to 0}

x = 6 or x = -2 {added 6 and subtracted 2, respectively}

Since a < b

a = -2 and b = 6

The minimum, is the y-coordinate of the vertex, -48. To find the the value of the x-coordinate of the

vertex, c, substitute -48 in for y into the equation y = 3x² - 12x - 36 and solve for x.

-48 = 3x² - 12x - 36

3x² - 12x + 12 = 0 {added 48 to each side}

3(x² - 4x + 4) = 0 {factored out the greatest common factor, 3}

x² - 4x + 4 = 0 {divided each side by 3}

(x - 2)(x - 2) = 0 {factored into two binomials}

x - 2 = 0 {set the factor equal to 0}

x = 2 {added 2 to each side}

Therefore, the x-coordinate of the vertex, c, is 2.

**B.) a = -2, b = 6, and c = 2**__Ask Algebra House__