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+7x+6> = 0

Set up the a, b, and c values:

a = 1, b = 7, c = 6

Quadratic Formula

x  =  -b ± √b2 - 4ac
  2a

Calculate -b

-b = -(7)

-b = -7

Calculate the discriminant Δ

Δ = b2 - 4ac:

Δ = 72 - 4 x 1 x 6

Δ = 49 - 24

Δ = 25 <--- Discriminant

Since Δ > 0, we expect two real roots.

Take the square root of Δ

Δ = √(25)

Δ = 5

-b + Δ:

Numerator 1 = -b + √Δ

Numerator 1 = -7 + 5

Numerator 1 = -2

-b - Δ:

Numerator 2 = -b - √Δ

Numerator 2 = -7 - 5

Numerator 2 = -12

Calculate 2a

Denominator = 2 * a

Denominator = 2 * 1

Denominator = 2

Find Solutions

Solution 1  =  Numerator 1
  Denominator

Solution 1  =  -2
  2

Solution 1 = -1

Solution 2

Solution 2  =  Numerator 2
  Denominator

Solution 2  =  -12
  2

Solution 2 = -6

Solution Set

(Solution 1, Solution 2) = (-1, -6)


Prove our first answer

(-1)2 + 7(-1) + 6 ? 0

(1) - 76 ? 0

1 - 76 ? 0

0 = 0

Prove our second answer

(-6)2 + 7(-6) + 6 ? 0

(36) - 426 ? 0

36 - 426 ? 0

0 = 0

(Solution 1, Solution 2) = (-1, -6)

Calculate the y-intercept

The y-intercept is the point where x = 0

Set x = 0 in ƒ(x) = x2 + 7x + 6 ≥

ƒ(0) = (0)2 + 7(0) + 6 ≥

ƒ(0) = 0 + 0 + 6

ƒ(0) = 6  ← Y-Intercept

Y-intercept = (0,6)

Vertex of a parabola

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

Use the formula rule.

Our equation coefficients are a = 1, b = 7

The formula rule determines h

h = Axis of Symmetry

h  =  -b
  2a

Plug in -b = -7 and a = 1

h  =  -(7)
  2(1)

h  =  -7
  2

h = -3.5  ← Axis of Symmetry

Calculate k

k = ƒ(h) where h = -3.5

ƒ(h) = (h)2(h)6 ≥

ƒ(-3.5) = (-3.5)2(-3.5)6 ≥

ƒ(-3.5) = 12.25 - 24.5 + 6

ƒ(-3.5) = -6.25

Our vertex (h,k) = (-3.5,-6.25)

Determine our vertex form:

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

Vertex form = (x + 3.5)2 - 6.25

Axis of Symmetry: h = -3.5
vertex (h,k) = (-3.5,-6.25)
Vertex form = (x + 3.5)2 - 6.25

Analyze the x2 coefficient

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

concave up

Subtract 6 to each side

x2 + 7x + 6 ≥ - 6 = 0 - 6

x2 - 24.5x = -6

Complete the square:

Add an amount to both sides

x2 + 7x + ? = -6 + ?

Add (½*middle coefficient)2 to each side

Amount to add  =  (1 x 7)2
  (2 x 1)2

Amount to add  =  (7)2
  (2)2

Amount to add  =  49
  4

Amount to add = 49/4

Rewrite our perfect square equation:

x2 + 7 + (7/2)2 = -6 + (7/2)2

(x + 7/2)2 = -6 + 49/4

Simplify Right Side of the Equation:

LCM of 1 and 4 = 4

We multiply -6 by 4 ÷ 1 = 4 and 49 by 4 ÷ 4 = 1

Simplified Fraction  =  -6 x 4 + 49 x 1
  4

Simplified Fraction  =  -24 + 49
  4

Simplified Fraction  =  25
  4

Our fraction can be reduced down:
Using our GCF of 25 and 4 = 25

Reducing top and bottom by 25 we get
1/0.16

We set our left side = u

u2 = (x + 7/2)2

u has two solutions:

u = +√1/0.16

u = -√1/0.16

Replacing u, we get:

x + 7/2 = +1

x + 7/2 = -1

Subtract 7/2 from the both sides

x + 7/2 - 7/2 = +1/1 - 7/2

Simplify right side of the equation

LCM of 1 and 2 = 2

We multiply 1 by 2 ÷ 1 = 2 and -7 by 2 ÷ 2 = 1

Simplified Fraction  =  1 x 2 - 7 x 1
  2

Simplified Fraction  =  2 - 7
  2

Simplified Fraction  =  -5
  2

Answer 1 = -5/2

Subtract 7/2 from the both sides

x + 7/2 - 7/2 = -1/1 - 7/2

Simplify right side of the equation

LCM of 1 and 2 = 2

We multiply -1 by 2 ÷ 1 = 2 and -7 by 2 ÷ 2 = 1

Simplified Fraction  =  -1 x 2 - 7 x 1
  2

Simplified Fraction  =  -2 - 7
  2

Simplified Fraction  =  -9
  2

Answer 2 = -9/2

Build factor pairs:

Since a = 1, find all factor pairs of c = 6
These must have a sum = 7

Factor Pairs of 6Sum of Factor Pair
-1,-6-1 - 6 = -7
-2,-3-2 - 3 = -5
6,16 + 1 = 7
3,23 + 2 = 5

We want {6,1}

Since our a coefficient = 1, we setup our factors

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

Factor: (x + 6)(x + 1)

Final Answer

(Solution 1, Solution 2) = (-1, -6)
Y-intercept = (0,6)
Axis of Symmetry: h = -3.5
vertex (h,k) = (-3.5,-6.25)
Vertex form = (x + 3.5)2 - 6.25
concave up
Factor: (x + 6)(x + 1)
Factor: (x + 6)(x + 1)





What is the Answer?
(Solution 1, Solution 2) = (-1, -6)
Y-intercept = (0,6)
Axis of Symmetry: h = -3.5
vertex (h,k) = (-3.5,-6.25)
Vertex form = (x + 3.5)2 - 6.25
concave up
Factor: (x + 6)(x + 1)
Factor: (x + 6)(x + 1)
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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