Solve, factor, complte the square
find the concavity, vertex, vertex form
axis of symmetry and y-intercept for the quadratic:
x2+2x = 35
The quadratic you entered is not in standard form:
ax2 + bx + c = 0
Subtract 35 from both sides
x2+2x - 35 = 35 - 35x2+2x - 35 = 0
Set up the a, b, and c values:
a = 1, b = 2, c = -35
Quadratic Formula
| x = | -b ± √b2 - 4ac |
| 2a |
Calculate -b
-b = -(2)
-b = -2
Calculate the discriminant Δ
Δ = b2 - 4ac:
Δ = 22 - 4 x 1 x -35
Δ = 4 - -140
Δ = 144 <--- Discriminant
Since Δ > 0, we expect two real roots.
Take the square root of Δ
√Δ = √(144)
√Δ = 12
-b + Δ:
Numerator 1 = -b + √Δ
Numerator 1 = -2 + 12
Numerator 1 = 10
-b - Δ:
Numerator 2 = -b - √Δ
Numerator 2 = -2 - 12
Numerator 2 = -14
Calculate 2a
Denominator = 2 * a
Denominator = 2 * 1
Denominator = 2
Find Solutions
| Solution 1 = | Numerator 1 |
| Denominator |
| Solution 1 = | 10 |
| 2 |
Solution 1 = 5
Solution 2
| Solution 2 = | Numerator 2 |
| Denominator |
| Solution 2 = | -14 |
| 2 |
Solution 2 = -7
Solution Set
(Solution 1, Solution 2) = (5, -7)
Prove our first answer
(5)2 + 2(5) - 35 ? 0
(25) + 1035 ? 0
25 + 1035 ? 0
0 = 0
Prove our second answer
(-7)2 + 2(-7) - 35 ? 0
(49) - 1435 ? 0
49 - 1435 ? 0
0 = 0
(Solution 1, Solution 2) = (5, -7)
Calculate the y-intercept
The y-intercept is the point where x = 0
Set x = 0 in ƒ(x) = x2 + 2x - 35=
ƒ(0) = (0)2 + 2(0) - 35=
ƒ(0) = 0 + 0 - 35
ƒ(0) = -35 ← Y-Intercept
Y-intercept = (0,-35)
Vertex of a parabola
(h,k) where y = a(x - h)2 + k
Use the formula rule.
Our equation coefficients are a = 1, b = 2
The formula rule determines h
h = Axis of Symmetry
| h = | -b |
| 2a |
Plug in -b = -2 and a = 1
| h = | -(2) |
| 2(1) |
| h = | -2 |
| 2 |
h = -1 ← Axis of Symmetry
Calculate k
k = ƒ(h) where h = -1
ƒ(h) = (h)2(h)35=
ƒ(-1) = (-1)2(-1)35=
ƒ(-1) = 1 - 2 - 35
ƒ(-1) = -36
Our vertex (h,k) = (-1,-36)
Determine our vertex form:
The vertex form is: a(x - h)2 + k
Vertex form = (x + 1)2 - 36
Axis of Symmetry: h = -1
vertex (h,k) = (-1,-36)
Vertex form = (x + 1)2 - 36
Analyze the x2 coefficient
Since our x2 coefficient of 1 is positive
The parabola formed by the quadratic is concave up
concave up
Add 35 to each side
x2 + 2x - 35= + 35 = 0 + 35
x2 - 2x = 35
Complete the square:
Add an amount to both sides
x2 + 2x + ? = 35 + ?
Add (½*middle coefficient)2 to each side
| Amount to add = | (1 x 2)2 |
| (2 x 1)2 |
| Amount to add = | (2)2 |
| (2)2 |
| Amount to add = | 4 |
| 4 |
Amount to add = 4/4
Rewrite our perfect square equation:
x2 + 2 + (2/2)2 = 35 + (2/2)2
(x + 2/2)2 = 35 + 4/4
Simplify Right Side of the Equation:
We multiply 35 by 4 ÷ 1 = 4 and 4 by 4 ÷ 4 = 1
| Simplified Fraction = | 35 x 4 + 4 x 1 |
| 4 |
| Simplified Fraction = | 140 + 4 |
| 4 |
| Simplified Fraction = | 144 |
| 4 |
Simplified Fraction = 36
We set our left side = u
u2 = (x + 2/2)2
u has two solutions:
u = +√36
u = -√36
Replacing u, we get:
x + 2/2 = +6
x + 2/2 = -6
Subtract 2/2 from the both sides
x + 2/2 - 2/2 = +6/1 - 2/2
Simplify right side of the equation
We multiply 6 by 2 ÷ 1 = 2 and -2 by 2 ÷ 2 = 1
| Simplified Fraction = | 6 x 2 - 2 x 1 |
| 2 |
| Simplified Fraction = | 12 - 2 |
| 2 |
| Simplified Fraction = | 10 |
| 2 |
Simplified Fraction = 5
Answer 1 = 5
Subtract 2/2 from the both sides
x + 2/2 - 2/2 = -6/1 - 2/2
Simplify right side of the equation
We multiply -6 by 2 ÷ 1 = 2 and -2 by 2 ÷ 2 = 1
| Simplified Fraction = | -6 x 2 - 2 x 1 |
| 2 |
| Simplified Fraction = | -12 - 2 |
| 2 |
| Simplified Fraction = | -14 |
| 2 |
Simplified Fraction = -7
Answer 2 = -7
Build factor pairs:
Since a = 1, find all factor pairs of c = -35
These must have a sum = 2
| Factor Pairs of -35 | Sum of Factor Pair |
|---|---|
| 1,-35 | 1 - 35 = -34 |
| 5,-7 | 5 - 7 = -2 |
| 7,-5 | 7 - 5 = 2 |
| 35,-1 | 35 - 1 = 34 |
We want {7,-5}
Since our a coefficient = 1, we setup our factors
(x + Factor Pair Answer 1)(x + Factor Pair Answer 2)
Factor: (x + 7)(x - 5)
Final Answer
Y-intercept = (0,-35)
Axis of Symmetry: h = -1
vertex (h,k) = (-1,-36)
Vertex form = (x + 1)2 - 36
concave up
Factor: (x + 7)(x - 5)
Factor: (x + 7)(x - 5)
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,-35)
Axis of Symmetry: h = -1
vertex (h,k) = (-1,-36)
Vertex form = (x + 1)2 - 36
concave up
Factor: (x + 7)(x - 5)
Factor: (x + 7)(x - 5)
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
