Hint Number =

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 = 1

 a  = -b ± √b2 - 4ac 2a

-b = -(1)

-b = -1

##### Calculate the discriminant Δ

Δ = b2 - 4ac:

Δ = 12 - 4 x 1 x 1

Δ = 1 - 4

Δ = -3 <--- Discriminant

Since Δ < 0, we expect two complex roots.

##### Take the square root of Δ

Δ = √(-3)

Since the square root < 0, we have an irrational answer

##### -b + Δ:

Numerator 1 = -b + √Δ

Numerator 1 = -1 + √-3

##### -b - Δ:

Numerator 2 = -b - √Δ

Numerator 2 = -1 - √-3

##### Calculate 2a

Denominator = 2 * a

Denominator = 2 * 1

Denominator = 2

##### Find Solutions

 Solution 1  = Numerator 1 Denominator

Solution 1 = (-1 + √-3)/2

##### Solution 2

 Solution 2  = Numerator 2 Denominator

Solution 2 = (-1 - √-3)/2

(Solution 1, Solution 2) = ((-1 + √3i)/2, (-1 - √3i)/2)

(Solution 1, Solution 2) = ((-1 + √3i)/2, (-1 - √3i)/2)

##### Calculate the y-intercept

The y-intercept is the point where x = 0

Set a = 0 in ƒ(a) = a2 + a + 1

ƒ(0) = (0)2 + (0) + 1

ƒ(0) = 0 + 0 + 1

ƒ(0) = 1  ← Y-Intercept

Y-intercept = (0,1)

##### 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) = a2a1

ƒ(-0.5) = a2a1

ƒ(-0.5) = 0.25 - 0.5 + 1

ƒ(-0.5) = 0.75

Our vertex (h,k) = (-0.5,0.75)

##### Determine our vertex form:

The vertex form is: a(x - h)2 + k

Vertex form = (x + 0.5)2 + 0.75

Axis of Symmetry: h = -0.5
vertex (h,k) = (-0.5,0.75)
Vertex form = (x + 0.5)2 + 0.75

##### Analyze the a2 coefficient

Since our a2 coefficient of 1 is positive
The parabola formed by the quadratic is concave up

concave up

##### Subtract 1 to each side

a2 + a + 1 - 1 = 0 - 1

a2 - 0.5a = -1

##### Complete the square:

Add an amount to both sides

a2 + 1a + ? = -1 + ?

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

##### Rewrite our perfect square equation:

a2 + 1 + (1/2)2 = -1 + (1/2)2

(a + 1/2)2 = -1 + 1/4

##### Simplify Right Side of the Equation:

LCM of 1 and 4 = 4

We multiply -1 by 4 ÷ 1 = 4 and 1 by 4 ÷ 4 = 1

 Simplified Fraction  = -1 x 4 + 1 x 1 4

 Simplified Fraction  = -4 + 1 4

 Simplified Fraction  = -3 4

u2 = (a + 1/2)2

u = +√-3/4

u = -√-3/4

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

LCM of 1 and 2 = 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

##### Subtract 1/2 from the both sides

a + 1/2 - 1/2 = -√/1 - 1/2

##### Simplify right side of the equation

LCM of 1 and 2 = 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

##### Build factor pairs:

Since a = 1, find all factor pairs of c = 1
These must have a sum = 1

Factor Pairs of 1Sum of Factor Pair
-1,-1-1 - 1 = -2
1,11 + 1 = 2

Since no factor pairs exist = 1, this quadratic cannot be factored any more

##### Since our a coefficient = 1, we setup our factors

Factor: (a + 0)(a + 0)

(Solution 1, Solution 2) = ((-1 + √3i)/2, (-1 - √3i)/2)
Y-intercept = (0,1)
Axis of Symmetry: h = -0.5
vertex (h,k) = (-0.5,0.75)
Vertex form = (x + 0.5)2 + 0.75
concave up
Factor: (a + 0)(a + 0)
Factor: (a + 0)(a + 0)

(Solution 1, Solution 2) = ((-1 + √3i)/2, (-1 - √3i)/2)
Y-intercept = (0,1)
Axis of Symmetry: h = -0.5
vertex (h,k) = (-0.5,0.75)
Vertex form = (x + 0.5)2 + 0.75
concave up
Factor: (a + 0)(a + 0)
Factor: (a + 0)(a + 0)
##### How does the Quadratic Equations and Inequalities Calculator work?
Free Quadratic Equations and Inequalities Calculator - Solves for quadratic equations in the form ax2 + bx + c = 0. Also generates practice problems as well as hints for each problem.
* Solve using the quadratic formula and the discriminant Δ
* Complete the Square for 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?

y = ax2 + bx + c
(-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

For more math formulas, check out our Formula Dossier

### 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