Solve, factor, complte the square
find the concavity, vertex, vertex form
axis of symmetry and y-intercept for the quadratic:
a2-a+6
Set up the a, b, and c values:
a = 1, b = 1, c = 6
Quadratic Formula
| a = | -b ± √b2 - 4ac |
| 2a |
Calculate -b
-b = -(1)
-b = -1
Calculate the discriminant Δ
Δ = b2 - 4ac:
Δ = 12 - 4 x 1 x 6
Δ = 1 - 24
Δ = -23 <--- Discriminant
Since Δ < 0, we expect two complex roots.
Take the square root of Δ
√Δ = √(-23)
Since the square root < 0, we have an irrational answer
-b + Δ:
Numerator 1 = -b + √Δ
Numerator 1 = -1 + √-23
-b - Δ:
Numerator 2 = -b - √Δ
Numerator 2 = -1 - √-23
Calculate 2a
Denominator = 2 * a
Denominator = 2 * 1
Denominator = 2
Find Solutions
| Solution 1 = | Numerator 1 |
| Denominator |
Solution 1 = (-1 + √-23)/2
Solution 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)
Calculate the y-intercept
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)
Vertex of a parabola
(h,k) where y = a(x - h)2 + k
Use the formula rule.
Our equation coefficients are a = 1, b = 1
The formula rule determines h
h = Axis of Symmetry
| h = | -b |
| 2a |
Plug in -b = -1 and a = 1
| h = | -(1) |
| 2(1) |
| h = | -1 |
| 2 |
h = -0.5 ← Axis of Symmetry
Calculate k
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)
Determine our vertex form:
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
Analyze the a2 coefficient
Since our a2 coefficient of 1 is positive
The parabola formed by the quadratic is concave up
concave up
Subtract 6 to each side
a2 + a + 6 - 6 = 0 - 6
a2 - 0.5a = -6
Complete the square:
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
Rewrite our perfect square equation:
a2 + 1 + (1/2)2 = -6 + (1/2)2
(a + 1/2)2 = -6 + 1/4
Simplify Right Side of the Equation:
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 |
We set our left side = u
u2 = (a + 1/2)2
u has two solutions:
u = +√-23/4
u = -√-23/4
Replacing u, we get:
a + 1/2 = +NAN
a + 1/2 = -NAN
Subtract 1/2 from the both sides
a + 1/2 - 1/2 = +√/1 - 1/2
Simplify right side of the equation
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
Subtract 1/2 from the both sides
a + 1/2 - 1/2 = -√/1 - 1/2
Simplify right side of the equation
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
Build factor pairs:
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
Since our a coefficient = 1, we setup our factors
(a + Factor Pair Answer 1)(a + Factor Pair Answer 2)
Factor: (a + 0)(a + 0)
Final Answer
Y-intercept = (0,6)
Axis of Symmetry: h = -0.5
vertex (h,k) = (-0.5,5.75)
Vertex form = (x + 0.5)2 + 5.75
concave up
Factor: (a + 0)(a + 0)
Factor: (a + 0)(a + 0)
Test Your Knowledge?
- What is the formula for a quadratic equation?___When does the quadratic parabola open concave up?___How are the discriminant and quadratic solution related?
Common Core State Standards In This Lesson
What is the Answer?
Y-intercept = (0,6)
Axis of Symmetry: h = -0.5
vertex (h,k) = (-0.5,5.75)
Vertex form = (x + 0.5)2 + 5.75
concave up
Factor: (a + 0)(a + 0)
Factor: (a + 0)(a + 0)
How does the Quadratic Equations and Inequalities Calculator work?
* Solve using the quadratic formula and the discriminant Δ
* Complete the Square for the Quadratic
* Factor the Quadratic
* Y-Intercept
* Vertex (h,k) of the parabola formed by the quadratic where h is the Axis of Symmetry as well as the vertex form of the equation a(h - h)2 + k
* Concavity of the parabola formed by the quadratic
* Using the Rational Root Theorem (Rational Zero Theorem), the calculator will determine potential roots which can then be tested against the synthetic calculator.
This calculator has 4 inputs.
What 5 formulas are used for the Quadratic Equations and Inequalities Calculator?
(-b ± √b2 - 4ac)/2a
h (Axis of Symmetry) = -b/2a
The vertex of a parabola is (h,k) where y = a(x - h)2 + k
What 9 concepts are covered in the Quadratic Equations and Inequalities Calculator?
- complete the square
- a technique for converting a quadratic polynomial of the form ax2 + bx + c to a(x - h)2 + k
- equation
- a statement declaring two mathematical expressions are equal
- factor
- a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n.
- intercept
- parabola
- a plane curve which is approximately U-shaped
- quadratic
- Polynomials with a maximum term degree as the second degree
- quadratic equations and inequalities
- rational root
- vertex
- Highest point or where 2 curves meet
