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:
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
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
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 6 | Sum of Factor Pair |
|---|---|
| -1,-6 | -1 - 6 = -7 |
| -2,-3 | -2 - 3 = -5 |
| 6,1 | 6 + 1 = 7 |
| 3,2 | 3 + 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
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)
Test Your Knowledge?
- What is the formula for a quadratic equation?___When does the quadratic parabola open concave up?___How are the discriminant and quadratic solution related?
Common Core State Standards In This Lesson
What is the Answer?
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?
* 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?
(-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
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
