Solve, factor, complte the square

find the concavity, vertex, vertex form

axis of symmetry and y-intercept for the quadratic:

x^{2}+2x = 35

The quadratic you entered is not in standard form:

ax^{2} + bx + c = 0

x

a = 1, b = 2, c = -35

x = | -b ± √b^{2} - 4ac |

2a |

-b = -(2)

-b = -2

Δ = b^{2} - 4ac:

Δ = 2^{2} - 4 x 1 x -35

Δ = 4 - -140

Δ = 144 <--- Discriminant

Since Δ > 0, we expect two real roots.

√Δ = √(144)

√Δ = 12

Numerator 1 = -b + √Δ

Numerator 1 = -2 + 12

Numerator 1 = 10

Numerator 2 = -b - √Δ

Numerator 2 = -2 - 12

Numerator 2 = -14

Denominator = 2 * a

Denominator = 2 * 1

Denominator = 2

Solution 1 = | Numerator 1 |

Denominator |

Solution 1 = | 10 |

2 |

Solution 1 = 5

Solution 2 = | Numerator 2 |

Denominator |

Solution 2 = | -14 |

2 |

Solution 2 = -7

(Solution 1, Solution 2) = (5, -7)

(5)^{2} + 2(5) - 35 ? 0

(25) + 1035 ? 0

25 + 1035 ? 0

0 = 0

(-7)^{2} + 2(-7) - 35 ? 0

(49) - 1435 ? 0

49 - 1435 ? 0

0 = 0

(Solution 1, Solution 2) = (5, -7)

The y-intercept is the point where x = 0

Set x = 0 in ƒ(x) = x^{2} + 2x - 35=

ƒ(0) = (0)^{2} + 2(0) - 35=

ƒ(0) = 0 + 0 - 35

ƒ(0) = **-35** ← Y-Intercept

Y-intercept = (0,-35)

(h,k) where y = a(x - h)^{2}+ k

Use the formula rule.

Our equation coefficients are a = 1, b = 2

h = Axis of Symmetry

h = | -b |

2a |

h = | -(2) |

2(1) |

h = | -2 |

2 |

**h = -1** ← Axis of Symmetry

k = ƒ(h) where h = -1

ƒ(h) = (h)^{2}(h)35=

ƒ(-1) = (-1)^{2}(-1)35=

ƒ(-1) = 1 - 2 - 35

ƒ(-1) = **-36**

Our vertex (**h,k**) = **(-1,-36)**

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

Vertex form = (x + 1)^{2} - 36

Axis of Symmetry: **h = -1**

vertex (**h,k**) = **(-1,-36)**

Vertex form = (x + 1)^{2} - 36

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

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

concave** up**

x^{2} + 2x - 35= + 35 = 0 + 35

x^{2} - 2x = 35

Add an amount to both sides

x^{2} + 2x + ? = 35 + ?

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

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

(2 x 1)^{2} |

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

(2)^{2} |

Amount to add = | 4 |

4 |

Amount to add = 4/4

x^{2} + 2 + (2/2)^{2} = 35 + (2/2)^{2}

(x + 2/2)^{2} = 35 + 4/4

We multiply 35 by 4 ÷ 1 = 4 and 4 by 4 ÷ 4 = 1

Simplified Fraction = | 35 x 4 + 4 x 1 |

4 |

Simplified Fraction = | 140 + 4 |

4 |

Simplified Fraction = | 144 |

4 |

Simplified Fraction = 36

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

u = +√36

u = -√36

x + 2/2 = +6

x + 2/2 = -6

x + 2/2 - 2/2 = +6/1 - 2/2

We multiply 6 by 2 ÷ 1 = 2 and -2 by 2 ÷ 2 = 1

Simplified Fraction = | 6 x 2 - 2 x 1 |

2 |

Simplified Fraction = | 12 - 2 |

2 |

Simplified Fraction = | 10 |

2 |

Simplified Fraction = 5

Answer 1 = **5**

x + 2/2 - 2/2 = -6/1 - 2/2

We multiply -6 by 2 ÷ 1 = 2 and -2 by 2 ÷ 2 = 1

Simplified Fraction = | -6 x 2 - 2 x 1 |

2 |

Simplified Fraction = | -12 - 2 |

2 |

Simplified Fraction = | -14 |

2 |

Simplified Fraction = -7

Answer 2 = **-7**

Since a = 1, find all factor pairs of c = -35

These must have a sum = 2

Factor Pairs of -35 | Sum of Factor Pair |
---|---|

1,-35 | 1 - 35 = -34 |

5,-7 | 5 - 7 = -2 |

7,-5 | 7 - 5 = 2 |

35,-1 | 35 - 1 = 34 |

We want {7,-5}

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

Factor: (x + 7)(x - 5)

(Solution 1, Solution 2) = (5, -7)

Y-intercept = (0,-35)

Axis of Symmetry:**h = -1**

vertex (**h,k**) = **(-1,-36)**

Vertex form = (x + 1)^{2} - 36

concave** up**

Factor: (x + 7)(x - 5)

Factor: (x + 7)(x - 5)

Y-intercept = (0,-35)

Axis of Symmetry:

vertex (

Vertex form = (x + 1)

concave

Factor: (x + 7)(x - 5)

Factor: (x + 7)(x - 5)

(Solution 1, Solution 2) = (5, -7)

Y-intercept = (0,-35)

Axis of Symmetry:**h = -1**

vertex (**h,k**) = **(-1,-36)**

Vertex form = (x + 1)^{2} - 36

concave** up**

Factor: (x + 7)(x - 5)

Factor: (x + 7)(x - 5)

Y-intercept = (0,-35)

Axis of Symmetry:

vertex (

Vertex form = (x + 1)

concave

Factor: (x + 7)(x - 5)

Factor: (x + 7)(x - 5)

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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