All for 4x^2+11x-3

April 23, 2024

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

Quadratic Formula

x =	-b ± √b2 - 4ac
	2a

Calculate -b

-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.

Take the square root of Δ

√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)

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

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

We set our left side = u

u2 = (x + 11/8)2

u has two solutions:

u = +√1/0.60952380952381

u = -√1/0.60952380952381

Replacing u, we get:

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

Answer 1 = -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

Answer 2 = -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 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
4x2( + 0x + 0x) + 1 = 0

Group terms

GCF of 4 and 0 = 1

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

( + 0)(x - 1) = 0

Final Answer

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

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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.
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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 squarea technique for converting a quadratic polynomial of the form ax2 + bx + c to a(x - h)2 + kequationa statement declaring two mathematical expressions are equalfactora 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.interceptparabolaa plane curve which is approximately U-shapedquadraticPolynomials with a maximum term degree as the second degreequadratic equations and inequalitiesrational rootvertexHighest point or where 2 curves meet

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