Solve, factor, complte the square

find the concavity, vertex, vertex form

axis of symmetry and y-intercept for the quadratic:

x^{2}+x-100>0

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

x = | -b ± √b^{2} - 4ac |

2a |

-b = -(1)

-b = -1

Δ = b^{2} - 4ac:

Δ = 1^{2} - 4 x 1 x -100

Δ = 1 - -400

Δ = 401 <--- Discriminant

Since Δ > 0, we expect two real roots.

√Δ = √(401)

√Δ = 1√401

Numerator 1 = -b + √Δ

Numerator 1 = -1 + 1√401

Numerator 2 = -b - √Δ

Numerator 2 = -1 - 1√401

Denominator = 2 * a

Denominator = 2 * 1

Denominator = 2

Solution 1 = | Numerator 1 |

Denominator |

Solution 1 =;(-1 + 1√401)/2

Solution 2 = | Numerator 2 |

Denominator |

Solution 2 = (-1 - 1√401)/2

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

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

The y-intercept is the point where x = 0

Set x = 0 in ƒ(x) = x^{2} + x - 100>

ƒ(0) = (0)^{2} + (0) - 100>

ƒ(0) = 0 + 0 - 100

ƒ(0) = **-100** ← Y-Intercept

Y-intercept = (0,-100)

(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) = (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)**

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

Since our x^{2} coefficient of 1 is positive

The parabola formed by the quadratic is concave** up**

concave** up**

x^{2} + x - 100> + 100 = 0 + 100

x^{2} - 0.5x = 100

Add an amount to both sides

x^{2} + 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 |

Amount to add = 1/4

x^{2} + 1 + (1/2)^{2} = 100 + (1/2)^{2}

(x + 1/2)^{2} = 100 + 1/4

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

u^{2} = (x + 1/2)^{2}

u = +√1/0.0099750623441397

u = -√1/0.0099750623441397

x + 1/2 = +1

x + 1/2 = -1

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

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 |

Answer 1 = **1/2**

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

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 |

Answer 2 = **-3/2**

Since a = 1, find all factor pairs of c = -100

These must have a sum = 1

Factor Pairs of -100 | Sum of Factor Pair |
---|---|

1,-100 | 1 - 100 = -99 |

2,-50 | 2 - 50 = -48 |

4,-25 | 4 - 25 = -21 |

5,-20 | 5 - 20 = -15 |

10,-10 | 10 - 10 = 0 |

20,-5 | 20 - 5 = 15 |

25,-4 | 25 - 4 = 21 |

50,-2 | 50 - 2 = 48 |

100,-1 | 100 - 1 = 99 |

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

(x + Factor Pair Answer 1)(x + Factor Pair Answer 2)

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)

Y-intercept = (0,-100)

Axis of Symmetry:

vertex (

Vertex form = (x + 0.5)

concave

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)

Y-intercept = (0,-100)

Axis of Symmetry:

vertex (

Vertex form = (x + 0.5)

concave

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

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

Free Quadratic Equations and Inequalities Calculator - Solves for quadratic equations in the form ax^{2} + 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

* 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.

* 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)

* 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.

y = ax^{2 + bx + c(-b ± √b2 - 4ac)/2ah (Axis of Symmetry) = -b/2aThe vertex of a parabola is (h,k) where y = a(x - h)2 + kFor more math formulas, check out our Formula Dossier}

- complete the square
- a technique for converting a quadratic polynomial of the form ax
^{2}+ 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

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