Solve, factor, complte the square

find the concavity, vertex, vertex form

axis of symmetry and y-intercept for the quadratic:

4x^{2}+11x-3

a = 4, b = 11, c = 1

x = | -b ± √b^{2} - 4ac |

2a |

-b = -(11)

-b = -11

Δ = b^{2} - 4ac:

Δ = 11^{2} - 4 x 4 x 1

Δ = 121 - 16

Δ = 105 <--- Discriminant

Since Δ > 0, we expect two real roots.

√Δ = √(105)

√Δ = 1√105

Numerator 1 = -b + √Δ

Numerator 1 = -11 + 1√105

Numerator 2 = -b - √Δ

Numerator 2 = -11 - 1√105

Denominator = 2 * a

Denominator = 2 * 4

Denominator = 8

Solution 1 = | Numerator 1 |

Denominator |

Solution 1 =;(-11 + 1√105)/8

Solution 2 = | Numerator 2 |

Denominator |

Solution 2 = (-11 - 1√105)/8

(Solution 1, Solution 2) = ((-11 + 1√105)/8, (-11 - 1√105)/8)

(Solution 1, Solution 2) = ((-11 + 1√105)/8, (-11 - 1√105)/8)

The y-intercept is the point where x = 0

Set x = 0 in ƒ(x) = 4x^{2} + 11x + 1

ƒ(0) = 4(0)^{2} + 11(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 = 4, b = 11

h = Axis of Symmetry

h = | -b |

2a |

h = | -(11) |

2(4) |

h = | -11 |

8 |

**h = -1.375** ← Axis of Symmetry

k = ƒ(h) where h = -1.375

ƒ(h) = (h)^{2}(h)1

ƒ(-1.375) = (-1.375)^{2}(-1.375)1

ƒ(-1.375) = 7.5625 - 15.125 + 1

ƒ(-1.375) = **-6.5625**

Our vertex (**h,k**) = **(-1.375,-6.5625)**

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

Vertex form = 4(x + 1.375)^{2} - 6.5625

Axis of Symmetry: **h = -1.375**

vertex (**h,k**) = **(-1.375,-6.5625)**

Vertex form = 4(x + 1.375)^{2} - 6.5625

Since our x^{2} coefficient of 4 is positive

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

concave** up**

4x^{2} + 11x + 1 - 1 = 0 - 1

4x^{2} - 15.125x = -1

Since our a coefficient of 4 ≠ 1

We divide our equation by 4

x^{2} + 11/4 = -1/4

Add an amount to both sides

x^{2} + 11/4x + ? = -1/4 + ?

Add (½*middle coefficient)^{2} to each side

Amount to add = | (1 x 11)^{2} |

(2 x 4)^{2} |

Amount to add = | (11)^{2} |

(8)^{2} |

Amount to add = | 121 |

64 |

Amount to add = 121/64

x^{2} + 11/4 + (11/8)^{2} = -1/4 + (11/8)^{2}

(x + 11/8)^{2} = -1/4 + 121/64

We multiply -1 by 64 ÷ 4 = 16 and 121 by 64 ÷ 64 = 1

Simplified Fraction = | -1 x 16 + 121 x 1 |

64 |

Simplified Fraction = | -16 + 121 |

64 |

Simplified Fraction = | 105 |

64 |

Our fraction can be reduced down:

Using our GCF of 105 and 64 = 105

Reducing top and bottom by 105 we get

1/0.60952380952381

u^{2} = (x + 11/8)^{2}

u = +√1/0.60952380952381

u = -√1/0.60952380952381

x + 11/8 = +1

x + 11/8 = -1

x + 11/8 - 11/8 = +1/1 - 11/8

We multiply 1 by 8 ÷ 1 = 8 and -11 by 8 ÷ 8 = 1

Simplified Fraction = | 1 x 8 - 11 x 1 |

8 |

Simplified Fraction = | 8 - 11 |

8 |

Simplified Fraction = | -3 |

8 |

Answer 1 = **-3/8**

x + 11/8 - 11/8 = -1/1 - 11/8

We multiply -1 by 8 ÷ 1 = 8 and -11 by 8 ÷ 8 = 1

Simplified Fraction = | -1 x 8 - 11 x 1 |

8 |

Simplified Fraction = | -8 - 11 |

8 |

Simplified Fraction = | -19 |

8 |

Answer 2 = **-19/8**

Since a = 4 ≠ 1, find all factor pairs:

a x c = 4 x 1 = 4

These must have a sum = 11

Factor Pairs of 4 | Sum of Factor Pair |
---|---|

-1,-4 | -1 - 4 = -5 |

-2,-2 | -2 - 2 = -4 |

4,1 | 4 + 1 = 5 |

2,2 | 2 + 2 = 4 |

Since no factor pairs exist = 11, this quadratic cannot be factored any more

Rewrite 11x as the sum of factor pairs:

0x + 0x

Our equation becomes

4x^{2}( + 0x + 0x) + 1 = 0

GCF of 4 and 0 = 1

GCF of 4 and 0 = 1

Factor out from the first group

Factor out from the second group

(x - 1) + (x + 0) = 0

Our common term is (x - 1)

Write this as ( + 0)(x - 1) = 0

If A x B = 0, then either A = 0 or B = 0

Set each factor to 0 and solve

Factor: ( + 0)(x - 1) = 0

(Solution 1, Solution 2) = ((-11 + 1√105)/8, (-11 - 1√105)/8)

Y-intercept = (0,1)

Axis of Symmetry:**h = -1.375**

vertex (**h,k**) = **(-1.375,-6.5625)**

Vertex form = 4(x + 1.375)^{2} - 6.5625

concave** up**

Factor: ( + 0)(x - 1) = 0

Factor: ( + 0)(x - 1) = 0

Y-intercept = (0,1)

Axis of Symmetry:

vertex (

Vertex form = 4(x + 1.375)

concave

Factor: ( + 0)(x - 1) = 0

Factor: ( + 0)(x - 1) = 0

(Solution 1, Solution 2) = ((-11 + 1√105)/8, (-11 - 1√105)/8)

Y-intercept = (0,1)

Axis of Symmetry:**h = -1.375**

vertex (**h,k**) = **(-1.375,-6.5625)**

Vertex form = 4(x + 1.375)^{2} - 6.5625

concave** up**

Factor: ( + 0)(x - 1) = 0

Factor: ( + 0)(x - 1) = 0

Y-intercept = (0,1)

Axis of Symmetry:

vertex (

Vertex form = 4(x + 1.375)

concave

Factor: ( + 0)(x - 1) = 0

Factor: ( + 0)(x - 1) = 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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