Hint Number =

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

4x2+11x-3

##### Set up the a, b, and c values:

a = 4, b = 11, c = 1

 x  = -b ± √b2 - 4ac 2a

-b = -(11)

-b = -11

##### Calculate the discriminant Δ

Δ = b2 - 4ac:

Δ = 112 - 4 x 4 x 1

Δ = 121 - 16

Δ = 105 <--- Discriminant

Since Δ > 0, we expect two real roots.

Δ = √(105)

Δ = 1√105

##### -b + Δ:

Numerator 1 = -b + √Δ

Numerator 1 = -11 + 1√105

##### -b - Δ:

Numerator 2 = -b - √Δ

Numerator 2 = -11 - 1√105

##### Calculate 2a

Denominator = 2 * a

Denominator = 2 * 4

Denominator = 8

##### Find Solutions

 Solution 1  = Numerator 1 Denominator

Solution 1  =;(-11 + 1√105)/8

##### Solution 2

 Solution 2  = Numerator 2 Denominator

Solution 2 = (-11 - 1√105)/8

##### Solution Set

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

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

##### Calculate the y-intercept

The y-intercept is the point where x = 0

Set x = 0 in ƒ(x) = 4x2 + 11x + 1

ƒ(0) = 4(0)2 + 11(0) + 1

ƒ(0) = 0 + 0 + 1

ƒ(0) = 1  ← Y-Intercept

Y-intercept = (0,1)

##### Vertex of a parabola

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

Use the formula rule.

Our equation coefficients are a = 4, b = 11

##### The formula rule determines h

h = Axis of Symmetry

 h  = -b 2a

##### Plug in -b = -11 and a = 4

 h  = -(11) 2(4)

 h  = -11 8

h = -1.375  ← Axis of Symmetry

##### Calculate k

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)

##### Determine our vertex form:

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

##### Analyze the x2 coefficient

Since our x2 coefficient of 4 is positive
The parabola formed by the quadratic is concave up

concave up

##### Subtract 1 to each side

4x2 + 11x + 1 - 1 = 0 - 1

4x2 - 15.125x = -1

Since our a coefficient of 4 ≠ 1
We divide our equation by 4

x2 + 11/4 = -1/4

##### Complete the square:

Add an amount to both sides

x2 + 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

##### Rewrite our perfect square equation:

x2 + 11/4 + (11/8)2 = -1/4 + (11/8)2

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

##### Simplify Right Side of the Equation:

LCM of 4 and 64 = 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

u2 = (x + 11/8)2

##### u has two solutions:

u = +√1/0.60952380952381

u = -√1/0.60952380952381

x + 11/8 = +1

x + 11/8 = -1

##### Subtract 11/8 from the both sides

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

##### Simplify right side of the equation

LCM of 1 and 8 = 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

##### Subtract 11/8 from the both sides

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

##### Simplify right side of the equation

LCM of 1 and 8 = 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

##### Build factor pairs:

Since a = 4 ≠ 1, find all factor pairs:
a x c = 4 x 1 = 4
These must have a sum = 11

Factor Pairs of 4Sum of Factor Pair
-1,-4-1 - 4 = -5
-2,-2-2 - 2 = -4
4,14 + 1 = 5
2,22 + 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
4x2( + 0x + 0x) + 1 = 0

##### Factor out terms:

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

##### Use the Zero Product Principal

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

(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
##### 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
* 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