Hint Number =

Solve, factor, complte the square
find the concavity, vertex, vertex form
axis of symmetry and y-intercept for the quadratic:

x2+x-100>0

##### Set up the a, b, and c values:

a = 1, b = 1, c = -100

 x  = -b ± √b2 - 4ac 2a

-b = -(1)

-b = -1

##### Calculate the discriminant Δ

Δ = b2 - 4ac:

Δ = 12 - 4 x 1 x -100

Δ = 1 - -400

Δ = 401 <--- Discriminant

Since Δ > 0, we expect two real roots.

Δ = √(401)

Δ = 1√401

##### -b + Δ:

Numerator 1 = -b + √Δ

Numerator 1 = -1 + 1√401

##### -b - Δ:

Numerator 2 = -b - √Δ

Numerator 2 = -1 - 1√401

##### Calculate 2a

Denominator = 2 * a

Denominator = 2 * 1

Denominator = 2

##### Find Solutions

 Solution 1  = Numerator 1 Denominator

Solution 1  =;(-1 + 1√401)/2

##### Solution 2

 Solution 2  = Numerator 2 Denominator

Solution 2 = (-1 - 1√401)/2

##### Solution Set

(Solution 1, Solution 2) = ((-1 + 1√401)/2, (-1 - 1√401)/2)

(Solution 1, Solution 2) = ((-1 + 1√401)/2, (-1 - 1√401)/2)

##### Calculate the y-intercept

The y-intercept is the point where x = 0

Set x = 0 in ƒ(x) = x2 + x - 100>

ƒ(0) = (0)2 + (0) - 100>

ƒ(0) = 0 + 0 - 100

ƒ(0) = -100  ← Y-Intercept

Y-intercept = (0,-100)

##### 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) = (h)2(h)100>

ƒ(-0.5) = (-0.5)2(-0.5)100>

ƒ(-0.5) = 0.25 - 0.5 - 100

ƒ(-0.5) = -100.25

Our vertex (h,k) = (-0.5,-100.25)

##### Determine our vertex form:

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

Vertex form = (x + 0.5)2 - 100.25

Axis of Symmetry: h = -0.5
vertex (h,k) = (-0.5,-100.25)
Vertex form = (x + 0.5)2 - 100.25

##### Analyze the x2 coefficient

Since our x2 coefficient of 1 is positive
The parabola formed by the quadratic is concave up

concave up

##### Add 100 to each side

x2 + x - 100> + 100 = 0 + 100

x2 - 0.5x = 100

##### Complete the square:

Add an amount to both sides

x2 + 1x + ? = 100 + ?

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:

x2 + 1 + (1/2)2 = 100 + (1/2)2

(x + 1/2)2 = 100 + 1/4

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

LCM of 1 and 4 = 4

We multiply 100 by 4 ÷ 1 = 4 and 1 by 4 ÷ 4 = 1

 Simplified Fraction  = 100 x 4 + 1 x 1 4

 Simplified Fraction  = 400 + 1 4

 Simplified Fraction  = 401 4

Our fraction can be reduced down:
Using our GCF of 401 and 4 = 401

Reducing top and bottom by 401 we get
1/0.0099750623441397

u2 = (x + 1/2)2

##### u has two solutions:

u = +√1/0.0099750623441397

u = -√1/0.0099750623441397

x + 1/2 = +1

x + 1/2 = -1

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

x + 1/2 - 1/2 = +1/1 - 1/2

##### Simplify right side of the equation

LCM of 1 and 2 = 2

We multiply 1 by 2 ÷ 1 = 2 and -1 by 2 ÷ 2 = 1

 Simplified Fraction  = 1 x 2 - 1 x 1 2

 Simplified Fraction  = 2 - 1 2

 Simplified Fraction  = 1 2

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

x + 1/2 - 1/2 = -1/1 - 1/2

##### Simplify right side of the equation

LCM of 1 and 2 = 2

We multiply -1 by 2 ÷ 1 = 2 and -1 by 2 ÷ 2 = 1

 Simplified Fraction  = -1 x 2 - 1 x 1 2

 Simplified Fraction  = -2 - 1 2

 Simplified Fraction  = -3 2

##### Build factor pairs:

Since a = 1, find all factor pairs of c = -100
These must have a sum = 1

Factor Pairs of -100Sum of Factor Pair
1,-1001 - 100 = -99
2,-502 - 50 = -48
4,-254 - 25 = -21
5,-205 - 20 = -15
10,-1010 - 10 = 0
20,-520 - 5 = 15
25,-425 - 4 = 21
50,-250 - 2 = 48
100,-1100 - 1 = 99

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

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

Factor: (x + 0)(x + 0)

(Solution 1, Solution 2) = ((-1 + 1√401)/2, (-1 - 1√401)/2)
Y-intercept = (0,-100)
Axis of Symmetry: h = -0.5
vertex (h,k) = (-0.5,-100.25)
Vertex form = (x + 0.5)2 - 100.25
concave up
Factor: (x + 0)(x + 0)
Factor: (x + 0)(x + 0)

(Solution 1, Solution 2) = ((-1 + 1√401)/2, (-1 - 1√401)/2)
Y-intercept = (0,-100)
Axis of Symmetry: h = -0.5
vertex (h,k) = (-0.5,-100.25)
Vertex form = (x + 0.5)2 - 100.25
concave up
Factor: (x + 0)(x + 0)
Factor: (x + 0)(x + 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