1)Find the exact solutions to 3x^2=5x-1 using the quadratic formula.

answer=5 plus or minus the square root of 37 over 6

2)Use the discriminant to determine the number and type of roots for the equation 2x^2-7x+9=0
answer=2 complex

3)Identify the vertex,axis of symmetry,and direction of opening for y=1/2(x-8)^2+2
answer=(8,2);x=-8;up

4)Which quadratic function has its vertex at(-2,7)and opens down?
answer=y=(x-2)^2+7

5)Write y=x^2+4x-1 in vertex form
answer=y=(x-2)^2+5

1)Find the exact solutions to 3x^2=5x-1 using the quadratic formula.
answer=5 plus or minus the square root of 37 over 6

2)Use the discriminant to determine the number and type of roots for the equation 2x^2-7x+9=0
answer=2 complex

3)Identify the vertex,axis of symmetry,and direction of opening for y=1/2(x-8)^2+2
answer=(8,2);x=-8;up

4)Which quadratic function has its vertex at(-2,7)and opens down?
answer=y=(x-2)^2+7

5)Write y=x^2+4x-1 in vertex form
answer=y=(x-2)^2+5

To find the exact solutions to the equation 3x^2 = 5x - 1 using the quadratic formula, we can use the formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 3, b = -5, and c = 1. Plugging in these values:

x = (-(-5) ± √((-5)^2 - 4(3)(-1))) / (2(3))
x = (5 ± √(25 + 12)) / 6
x = (5 ± √37) / 6

So the exact solutions to the equation are:

x = (5 + √37) / 6
x = (5 - √37) / 6

Using the discriminant to determine the number and type of roots for the equation 2x^2 - 7x + 9 = 0, we can use the formula for the discriminant:

D = b^2 - 4ac

In this case, a = 2, b = -7, and c = 9. Plugging in these values:

D = (-7)^2 - 4(2)(9)
D = 49 - 72
D = -23

The discriminant is negative (-23), which means there are two complex roots.

To identify the vertex, axis of symmetry, and direction of opening for the equation y = 1/2(x - 8)^2 + 2, we can use the vertex form of a quadratic equation:

y = a(x - h)^2 + k

In this case, the vertex form is y = 1/2(x - 8)^2 + 2. The vertex is at (h, k), so the vertex is (8, 2). The axis of symmetry is the vertical line x = h, so the axis of symmetry is x = 8. The direction of opening can be determined by the coefficient a. If a is positive, the parabola opens upward, and if a is negative, the parabola opens downward. In this case, a is positive (1/2), so the direction of opening is upward.

To find a quadratic function with vertex at (-2, 7) and opens down, we can use the vertex form of a quadratic equation:

y = a(x - h)^2 + k

In this case, the vertex is (-2, 7). Plugging in these values:

y = a(x - (-2))^2 + 7
y = a(x + 2)^2 + 7

Since the parabola opens down, a is negative. So the quadratic function is:

y = a(x + 2)^2 + 7

To write the equation y = x^2 + 4x - 1 in vertex form, we can start by completing the square. Rearrange the equation:

y = x^2 + 4x - 1

Now we want to find a value that we can add and subtract from the right side to create a perfect square trinomial. In this case, the value is (4/2)^2 = 4. Add and subtract 4 from the right side:

y = x^2 + 4x - 1 + 4 - 4

Simplify:

y = x^2 + 4x + 3 - 4

Factor the perfect square trinomial:

y = (x^2 + 4x + 4) - 4 + 3

Simplify:

y = (x + 2)^2 - 1

So the equation in vertex form is:

y = (x + 2)^2 - 1.