Enter Quadratic equation/inequality below

Hint Number =

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

x2+2x = 35

The quadratic you entered is not in standard form:
ax2 + bx + c = 0

Subtract 35 from both sides

x2+2x - 35 = 35 - 35
x2+2x - 35 = 0

Set up the a, b, and c values:

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

Quadratic Formula

x  =  -b ± √b2 - 4ac
  2a

Calculate -b

-b = -(2)

-b = -2

Calculate the discriminant Δ

Δ = b2 - 4ac:

Δ = 22 - 4 x 1 x -35

Δ = 4 - -140

Δ = 144 <--- Discriminant

Since Δ > 0, we expect two real roots.

Take the square root of Δ

Δ = √(144)

Δ = 12

-b + Δ:

Numerator 1 = -b + √Δ

Numerator 1 = -2 + 12

Numerator 1 = 10

-b - Δ:

Numerator 2 = -b - √Δ

Numerator 2 = -2 - 12

Numerator 2 = -14

Calculate 2a

Denominator = 2 * a

Denominator = 2 * 1

Denominator = 2

Find Solutions

Solution 1  =  Numerator 1
  Denominator

Solution 1  =  10
  2

Solution 1 = 5

Solution 2

Solution 2  =  Numerator 2
  Denominator

Solution 2  =  -14
  2

Solution 2 = -7

Solution Set

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


Prove our first answer

(5)2 + 2(5) - 35 ? 0

(25) + 1035 ? 0

25 + 1035 ? 0

0 = 0

Prove our second answer

(-7)2 + 2(-7) - 35 ? 0

(49) - 1435 ? 0

49 - 1435 ? 0

0 = 0

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

Calculate the y-intercept

The y-intercept is the point where x = 0

Set x = 0 in ƒ(x) = x2 + 2x - 35=

ƒ(0) = (0)2 + 2(0) - 35=

ƒ(0) = 0 + 0 - 35

ƒ(0) = -35  ← Y-Intercept

Y-intercept = (0,-35)

Vertex of a parabola

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

Use the formula rule.

Our equation coefficients are a = 1, b = 2

The formula rule determines h

h = Axis of Symmetry

h  =  -b
  2a

Plug in -b = -2 and a = 1

h  =  -(2)
  2(1)

h  =  -2
  2

h = -1  ← Axis of Symmetry

Calculate k

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)

Determine our vertex form:

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

Analyze the x2 coefficient

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

concave up

Add 35 to each side

x2 + 2x - 35= + 35 = 0 + 35

x2 - 2x = 35

Complete the square:

Add an amount to both sides

x2 + 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

Rewrite our perfect square equation:

x2 + 2 + (2/2)2 = 35 + (2/2)2

(x + 2/2)2 = 35 + 4/4

Simplify Right Side of the Equation:

LCM of 1 and 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

We set our left side = u

u2 = (x + 2/2)2

u has two solutions:

u = +√36

u = -√36

Replacing u, we get:

x + 2/2 = +6

x + 2/2 = -6

Subtract 2/2 from the both sides

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

Simplify right side of the equation

LCM of 1 and 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

Subtract 2/2 from the both sides

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

Simplify right side of the equation

LCM of 1 and 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

Build factor pairs:

Since a = 1, find all factor pairs of c = -35
These must have a sum = 2

Factor Pairs of -35Sum of Factor Pair
1,-351 - 35 = -34
5,-75 - 7 = -2
7,-57 - 5 = 2
35,-135 - 1 = 34

We want {7,-5}

Since our a coefficient = 1, we setup our factors

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

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

Final Answer

(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)





What is the Answer?
(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)
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
* 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.

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
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
Example calculations for the Quadratic Equations and Inequalities Calculator
Quadratic Equations and Inequalities Calculator Video

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