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, watch your signs
3x^2 - 5x + 1=0
x=(5±√(25-12))/6 = (5±√(13))/6
#2, ok
#3, vertex is ok, but how can the axis of symmetry be x=-8?? Would it not go through the vertex?? So it is x=9 (always the same as the x of the vertex)
#4 yours opens UP
it should be y = -(x-2)^2+5
#5 no, it should be y=(x+2)^2-5
(why don't you expand your answer, it would not work)
1) To solve the equation 3x^2 = 5x - 1 using the quadratic formula, we need to rearrange the equation in standard quadratic form: ax^2 + bx + c = 0. In this case, a = 3, b = -5, and c = -1. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a).
Substituting the values into the formula, we get:
x = (5 ± √((-5)^2 - 4 * 3 * (-1))) / (2 * 3)
x = (5 ± √(25 + 12)) / 6
x = (5 ± √37) / 6
So the exact solutions are x = (5 + √37) / 6 and x = (5 - √37) / 6.
2) The discriminant is used to determine the number and type of roots of a quadratic equation. For the equation 2x^2 - 7x + 9 = 0, the discriminant is given by b^2 - 4ac.
In this case, a = 2, b = -7, and c = 9. Substituting the values into the discriminant formula, we have:
b^2 - 4ac = (-7)^2 - 4 * 2 * 9
= 49 - 72
= -23
Since the discriminant is negative (-23), there are no real roots. Therefore, the equation has 2 complex roots.
3) The quadratic equation y = (1/2)(x - 8)^2 + 2 is already in vertex form. The vertex form of a quadratic equation is y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
The vertex of the equation is (8, 2). The axis of symmetry is the vertical line passing through the vertex, so it is x = 8. Since the coefficient of x^2 is positive (1/2), the parabola opens upwards.
4) To find a quadratic function with a vertex at (-2, 7) and opens downwards, we use the vertex form of a quadratic equation: y = a(x - h)^2 + k.
Since the vertex is (-2, 7), we have h = -2 and k = 7. The coefficient a determines whether the parabola opens upwards (a > 0) or downwards (a < 0). In this case, since the parabola opens downwards, a < 0.
Therefore, the quadratic function is y = a(x + 2)^2 + 7.
5) To write the equation y = x^2 + 4x - 1 in vertex form, we need to complete the square. The vertex form of a quadratic equation is y = a(x - h)^2 + k.
First, we group the terms with x:
y = (x^2 + 4x) - 1
Next, we complete the square by adding and subtracting (b/2)^2 to the expression inside the parentheses. In this case, b = 4, so we add and subtract (4/2)^2 = 4:
y = (x^2 + 4x + 4 - 4) - 1
We write the first three terms as a perfect square:
y = (x + 2)^2 - 4 - 1
Combine the constants:
y = (x + 2)^2 - 5
So the equation y = x^2 + 4x - 1 can be written as y = (x + 2)^2 - 5 in vertex form.