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 = -3
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 -3
Δ = 121 - -48
Δ = 169 <--- Discriminant
Since Δ > 0, we expect two real roots.
Take the square root of Δ
√Δ = √(169)
√Δ = 13
-b + Δ:
Numerator 1 = -b + √Δ
Numerator 1 = -11 + 13
Numerator 1 = 2
-b - Δ:
Numerator 2 = -b - √Δ
Numerator 2 = -11 - 13
Numerator 2 = -24
Calculate 2a
Denominator = 2 * a
Denominator = 2 * 4
Denominator = 8
Find Solutions
| Solution 1 = | Numerator 1 |
| Denominator |
| Solution 1 = | 2 |
| 8 |
Solution 1 = 0.25 or 1/4
Solution 2
| Solution 2 = | Numerator 2 |
| Denominator |
| Solution 2 = | -24 |
| 8 |
Solution 2 = -3
Solution Set
(Solution 1, Solution 2) = (0.25, -3)
Prove our first answer
(0.25)2 + 11(0.25) - 3 ? 0
(0.0625) + 2.753 ? 0
0.25 + 2.753 ? 0
0 = 0
Prove our second answer
(-3)2 + 11(-3) - 3 ? 0
(9) - 333 ? 0
36 - 333 ? 0
0 = 0
(Solution 1, Solution 2) = (0.25, -3)
Calculate the y-intercept
The y-intercept is the point where x = 0
Set x = 0 in ƒ(x) = 4x2 + 11x - 3
ƒ(0) = 4(0)2 + 11(0) - 3
ƒ(0) = 0 + 0 - 3
ƒ(0) = -3 ← Y-Intercept
Y-intercept = (0,-3)
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)3
ƒ(-1.375) = (-1.375)2(-1.375)3
ƒ(-1.375) = 7.5625 - 15.125 - 3
ƒ(-1.375) = -10.5625
Our vertex (h,k) = (-1.375,-10.5625)
Determine our vertex form:
The vertex form is: a(x - h)2 + k
Vertex form = 4(x + 1.375)2 - 10.5625
Axis of Symmetry: h = -1.375
vertex (h,k) = (-1.375,-10.5625)
Vertex form = 4(x + 1.375)2 - 10.5625
Analyze the x2 coefficient
Since our x2 coefficient of 4 is positive
The parabola formed by the quadratic is concave up
concave up
Add 3 to each side
4x2 + 11x - 3 + 3 = 0 + 3
4x2 - 15.125x = 3
Since our a coefficient of 4 ≠ 1
We divide our equation by 4
x2 + 11/4 = 3/4
Complete the square:
Add an amount to both sides
x2 + 11/4x + ? = 3/4 + ?
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 = 3/4 + (11/8)2
(x + 11/8)2 = 3/4 + 121/64
Simplify Right Side of the Equation:
We multiply 3 by 64 ÷ 4 = 16 and 121 by 64 ÷ 64 = 1
| Simplified Fraction = | 3 x 16 + 121 x 1 |
| 64 |
| Simplified Fraction = | 48 + 121 |
| 64 |
| Simplified Fraction = | 169 |
| 64 |
Our fraction can be reduced down:
Using our GCF of 169 and 64 = 169
Reducing top and bottom by 169 we get
1/0.37869822485207
We set our left side = u
u2 = (x + 11/8)2
u has two solutions:
u = +√1/0.37869822485207
u = -√1/0.37869822485207
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
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
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 -3 = -12
These must have a sum = 11
| Factor Pairs of -12 | Sum of Factor Pair |
|---|---|
| 1,-12 | 1 - 12 = -11 |
| 2,-6 | 2 - 6 = -4 |
| 3,-4 | 3 - 4 = -1 |
| 4,-3 | 4 - 3 = 1 |
| 6,-2 | 6 - 2 = 4 |
| 12,-1 | 12 - 1 = 11 |
We want {12,-1}
Rewrite 11x as the sum of factor pairs:
12x - 1x
Our equation becomes
4x2( + 12x - 1x) - 3 = 0
Group terms
GCF of 4 and 12 = 4
GCF of 4 and -1 = 1
First GCF
Our first GCF is an integer
Group the terms below:
4x2 and 12x and -1x and -3
This can be written as:
(4x2 + 12x) + (-1x - 3) = 0
Factor out terms:
Factor out 4x from the first group
Factor out 1 from the second group
4x(x + 3) + 1(-x - 3) = 0
Our common term is (x + 3)
Write this as (4x + 1)(x + 3) = 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: (4x + 1)(x + 3) = 0
Final Answer
Y-intercept = (0,-3)
Axis of Symmetry: h = -1.375
vertex (h,k) = (-1.375,-10.5625)
Vertex form = 4(x + 1.375)2 - 10.5625
concave up
Factor: (4x + 1)(x + 3) = 0
Factor: (4x + 1)(x + 3) = 0
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,-3)
Axis of Symmetry: h = -1.375
vertex (h,k) = (-1.375,-10.5625)
Vertex form = 4(x + 1.375)2 - 10.5625
concave up
Factor: (4x + 1)(x + 3) = 0
Factor: (4x + 1)(x + 3) = 0
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
