Question

1. what is the value of the discriminant of the equation $x^2 - 3x + 4 = 0$? then, use the value to determine whether the equation has two real solutions, one real solution, or no real solutions (two imaginary solutions).

a. 0; one real solution
b. -7; no real solution
c. 9; two real solutions
d. 8; two imaginary solutions

1. what is the vertex of the following function?

$y = 2(x - 4)^2 + 3$
a. (4,3)
b. (4,-3)
c. (8,6)
d. (-4,3)

1. determine the vertex form of $y = x^2 + 8x + 10$.

a. $y = (x + 4)^2 + 10$
b. $y = (x + 4)^2 - 6$
c. $y = (x + 4)^2 + 26$
d. $y = (x - 4)^2 - 6$

1. what is the the axis of symmetry of the quadratic equation $y = 4x^2 + 6x - 12$?

a. $x = -12$
b. $x = \frac{1}{2}$
c. $x = \frac{3}{4}$
d. $x = -\frac{3}{4}$

1. what is the axis of symmetry of the following equation?

$y = (x - 1)(x + 5)$
a. $x = -2$
b. $x = 1$
c. $x = -5$
d. $x = 2$

1. factor the following expression:

$x^2 - x - 6$

Explanation:

Question 5

Step1: Recall discriminant formula

For a quadratic equation \(ax^2 + bx + c = 0\), the discriminant \(D = b^2 - 4ac\). Here, \(a = 1\), \(b=-3\), \(c = 4\).

Step2: Calculate discriminant

\(D=(-3)^2-4\times1\times4 = 9 - 16=-7\).

Step3: Determine solution type

Since \(D<0\), the equation has no real solutions (two imaginary solutions).

Step1: Recall vertex form of parabola

The vertex form of a parabola is \(y = a(x - h)^2 + k\), where \((h,k)\) is the vertex.

Step2: Identify h and k

For \(y = 2(x - 4)^2 + 3\), \(h = 4\) and \(k = 3\). So the vertex is \((4,3)\).

Step1: Complete the square

For \(y=x^2 + 8x + 10\), take coefficient of \(x\) which is 8, half of it is 4, square it: \(4^2 = 16\).

Step2: Rewrite the equation

\(y=(x^2 + 8x + 16)-16 + 10=(x + 4)^2-6\).

Answer:

b. -7; no real solution

Question 6