Solve, factor, complte the square

find the concavity, vertex, vertex form

axis of symmetry and y-intercept for the quadratic:

a^{2}-a+6

a = 1, b = 1, c = 1

a = | -b ± √b^{2} - 4ac |

2a |

-b = -(1)

-b = -1

Δ = b^{2} - 4ac:

Δ = 1^{2} - 4 x 1 x 1

Δ = 1 - 4

Δ = -3 <--- Discriminant

Since Δ < 0, we expect two complex roots.

√Δ = √(-3)

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

Numerator 1 = -b + √Δ

Numerator 1 = -1 + √-3

Numerator 2 = -b - √Δ

Numerator 2 = -1 - √-3

Denominator = 2 * a

Denominator = 2 * 1

Denominator = 2

Solution 1 = | Numerator 1 |

Denominator |

Solution 1 = (-1 + √-3)/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)

The y-intercept is the point where x = 0

Set a = 0 in ƒ(a) = a^{2} + a + 1

ƒ(0) = (0)^{2} + (0) + 1

ƒ(0) = 0 + 0 + 1

ƒ(0) = **1** ← Y-Intercept

Y-intercept = (0,1)

(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) = a^{2}a1

ƒ(-0.5) = a^{2}a1

ƒ(-0.5) = 0.25 - 0.5 + 1

ƒ(-0.5) = **0.75**

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

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

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

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

concave** up**

a^{2} + a + 1 - 1 = 0 - 1

a^{2} - 0.5a = -1

Add an amount to both sides

a^{2} + 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 |

Amount to add = 1/4

a^{2} + 1 + (1/2)^{2} = -1 + (1/2)^{2}

(a + 1/2)^{2} = -1 + 1/4

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 |

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

u = +√-3/4

u = -√-3/4

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 = 1

These must have a sum = 1

Factor Pairs of 1 | Sum of Factor Pair |
---|---|

-1,-1 | -1 - 1 = -2 |

1,1 | 1 + 1 = 2 |

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)

(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)

Y-intercept = (0,1)

Axis of Symmetry:

vertex (

Vertex form = (x + 0.5)

concave

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)

Y-intercept = (0,1)

Axis of Symmetry:

vertex (

Vertex form = (x + 0.5)

concave

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

Factor: (a + 0)(a + 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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