l
Solve, factor, complte the square
find the concavity, vertex, vertex form
axis of symmetry and y-intercept for the quadratic:
a2-a+6
a = 1, b = 1, c = 6
a = | -b ± √b2 - 4ac |
2a |
-b = -(1)
-b = -1
Δ = b2 - 4ac:
Δ = 12 - 4 x 1 x 6
Δ = 1 - 24
Δ = -23 <--- Discriminant
Since Δ < 0, we expect two complex roots.
√Δ = √(-23)
Since the square root < 0, we have an irrational answer
Numerator 1 = -b + √Δ
Numerator 1 = -1 + √-23
Numerator 2 = -b - √Δ
Numerator 2 = -1 - √-23
Denominator = 2 * a
Denominator = 2 * 1
Denominator = 2
Solution 1 = | Numerator 1 |
Denominator |
Solution 1 = (-1 + √-23)/2
Solution 2 = | Numerator 2 |
Denominator |
Solution 2 = (-1 - √-23)/2
(Solution 1, Solution 2) = ((-1 + √23i)/2, (-1 - √23i)/2)
(Solution 1, Solution 2) = ((-1 + √23i)/2, (-1 - √23i)/2)
The y-intercept is the point where x = 0
Set a = 0 in ƒ(a) = a2 + a + 6
ƒ(0) = (0)2 + (0) + 6
ƒ(0) = 0 + 0 + 6
ƒ(0) = 6 ← Y-Intercept
Y-intercept = (0,6)
(h,k) where y = a(x - h)2 + k
Use the formula rule.
Our equation coefficients are a = 1, b = 1
h = Axis of Symmetry
h = | -b |
2a |
h = | -(1) |
2(1) |
h = | -1 |
2 |
h = -0.5 ← Axis of Symmetry
k = ƒ(h) where h = -0.5
ƒ(h) = a2a6
ƒ(-0.5) = a2a6
ƒ(-0.5) = 0.25 - 0.5 + 6
ƒ(-0.5) = 5.75
Our vertex (h,k) = (-0.5,5.75)
The vertex form is: a(x - h)2 + k
Vertex form = (x + 0.5)2 + 5.75
Axis of Symmetry: h = -0.5
vertex (h,k) = (-0.5,5.75)
Vertex form = (x + 0.5)2 + 5.75
Since our a2 coefficient of 1 is positive
The parabola formed by the quadratic is concave up
concave up
a2 + a + 6 - 6 = 0 - 6
a2 - 0.5a = -6
Add an amount to both sides
a2 + 1a + ? = -6 + ?
Add (½*middle coefficient)2 to each side
Amount to add = | (1 x 1)2 |
(2 x 1)2 |
Amount to add = | (1)2 |
(2)2 |
Amount to add = | 1 |
4 |
Amount to add = 1/4
a2 + 1 + (1/2)2 = -6 + (1/2)2
(a + 1/2)2 = -6 + 1/4
We multiply -6 by 4 ÷ 1 = 4 and 1 by 4 ÷ 4 = 1
Simplified Fraction = | -6 x 4 + 1 x 1 |
4 |
Simplified Fraction = | -24 + 1 |
4 |
Simplified Fraction = | -23 |
4 |
u2 = (a + 1/2)2
u = +√-23/4
u = -√-23/4
a + 1/2 = +NAN
a + 1/2 = -NAN
a + 1/2 - 1/2 = +√/1 - 1/2
We multiply √ by 2 ÷ 1 = 2 and -1 by 2 ÷ 2 = 1
Simplified Fraction = | √ x 2 - 1 x 1 |
2 |
Simplified Fraction = | 0 - 1 |
2 |
Simplified Fraction = | -1 |
2 |
Answer 1 = -1/2
a + 1/2 - 1/2 = -√/1 - 1/2
We multiply 0 by 2 ÷ 1 = 2 and -1 by 2 ÷ 2 = 1
Simplified Fraction = | 0 x 2 - 1 x 1 |
2 |
Simplified Fraction = | 0 - 1 |
2 |
Simplified Fraction = | -1 |
2 |
Answer 2 = -1/2
Since a = 1, find all factor pairs of c = 6
These must have a sum = 1
Factor Pairs of 6 | Sum of Factor Pair |
---|---|
-1,-6 | -1 - 6 = -7 |
-2,-3 | -2 - 3 = -5 |
6,1 | 6 + 1 = 7 |
3,2 | 3 + 2 = 5 |
Since no factor pairs exist = 1, this quadratic cannot be factored any more
(a + Factor Pair Answer 1)(a + Factor Pair Answer 2)
Factor: (a + 0)(a + 0)