Solve, factor, complte the square find the concavity, vertex, vertex form axis of symmetry and y-intercept for the quadratic:
a2 -a+6
Set up the a , b , and c values: a = 1 , b = 1 , c = 6
Quadratic Formula
Calculate -b -b = -(1)
-b = -1
Calculate the discriminant Δ Δ = b2 - 4ac:
Δ = 12 - 4 x 1 x 6
Δ = 1 - 24
Δ = -23 <--- Discriminant
Since Δ < 0, we expect two complex roots.
Take the square root of Δ √Δ = √(-23)
Since the square root < 0, we have an irrational answer
-b + Δ: Numerator 1 = -b + √Δ
Numerator 1 = -1 + √-23
-b - Δ: Numerator 2 = -b - √Δ
Numerator 2 = -1 - √-23
Calculate 2a Denominator = 2 * a
Denominator = 2 * 1
Denominator = 2
Find Solutions Solution 1 = Numerator 1 Denominator
Solution 1 = (-1 + √-23 )/2
Solution 2 Solution 2 = Numerator 2 Denominator
Solution 2 = (-1 - √-23 )/2
(Solution 1, Solution 2) = ((-1 + √23 i)/2, (-1 - √23 i)/2)
(Solution 1, Solution 2) = ((-1 + √23 i)/2, (-1 - √23 i)/2)
Calculate the y-intercept
The y-intercept is the point where x = 0 Set a = 0 in ƒ(a) = a2 + a + 6
ƒ(0 ) = (0 )2 + (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 = 1
The formula rule determines h h = Axis of Symmetry
Plug in -b = -1 and a = 1 h = -0.5 ← Axis of Symmetry
Calculate k k = ƒ(h ) where h = -0.5
ƒ(h ) = a2 a6
ƒ(-0.5 ) = a2 a6
ƒ(-0.5 ) = 0.25 - 0.5 + 6
ƒ(-0.5 ) = 5.75
Our vertex (h ,k ) = (-0.5 ,5.75)
Determine our vertex form: The vertex form is: a(x - h)2 + k
Vertex form = (x + 0.5)2 + 5.75
Axis of Symmetry: h = -0.5 vertex (h ,k ) = (-0.5 ,5.75) Vertex form = (x + 0.5)2 + 5.75
Analyze the a2 coefficient Since our a2 coefficient of 1 is positive The parabola formed by the quadratic is concave up
concave up
Subtract 6 to each side a2 + a + 6 - 6 = 0 - 6
a2 - 0.5a = -6
Complete the square: Add an amount to both sides
a2 + 1a + ? = -6 + ?
Add (½*middle coefficient)2 to each side
Amount to add = (1 x 1)2 (2 x 1)2
Amount to add = (1)2 (2)2
Amount to add = 1/4
Rewrite our perfect square equation: a2 + 1 + (1/2)2 = -6 + (1/2)2
(a + 1/2)2 = -6 + 1/4
Simplify Right Side of the Equation: LCM of 1 and 4 = 4
We multiply -6 by 4 ÷ 1 = 4 and 1 by 4 ÷ 4 = 1
Simplified Fraction = -6 x 4 + 1 x 1 4
Simplified Fraction = -24 + 1 4
Simplified Fraction = -23 4
We set our left side = u u2 = (a + 1/2)2
u has two solutions: u = +√-23/4
u = -√-23/4
Replacing u, we get: a + 1/2 = +NAN
a + 1/2 = -NAN
Subtract 1/2 from the both sides a + 1/2 - 1/2 = +√ /1 - 1/2
Simplify right side of the equation LCM of 1 and 2 = 2
We multiply √ by 2 ÷ 1 = 2 and -1 by 2 ÷ 2 = 1
Simplified Fraction = √ x 2 - 1 x 1 2
Simplified Fraction = 0 - 1 2
Simplified Fraction = -1 2
Answer 1 = -1/2
Subtract 1/2 from the both sides a + 1/2 - 1/2 = -√ /1 - 1/2
Simplify right side of the equation LCM of 1 and 2 = 2
We multiply 0 by 2 ÷ 1 = 2 and -1 by 2 ÷ 2 = 1
Simplified Fraction = 0 x 2 - 1 x 1 2
Simplified Fraction = 0 - 1 2
Simplified Fraction = -1 2
Answer 2 = -1/2
Build factor pairs: Since a = 1, find all factor pairs of c = 6 These must have a sum = 1
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
Since no factor pairs exist = 1, this quadratic cannot be factored any more
Since our a coefficient = 1, we setup our factors (a + Factor Pair Answer 1)(a + Factor Pair Answer 2)
Factor: (a + 0)(a + 0)
Final Answer (Solution 1, Solution 2) = ((-1 + √23 i)/2, (-1 - √23 i)/2) Y-intercept = (0,6) Axis of Symmetry: h = -0.5 vertex (h ,k ) = (-0.5 ,5.75) Vertex form = (x + 0.5)2 + 5.75 concave up Factor: (a + 0)(a + 0) Factor: (a + 0)(a + 0)
Common Core State Standards In This Lesson
HSN.CN.C.7, HSA.SSE.B.3.A, HSA.SSE.B.3.B, HSA.REI.B.4, HSA.REI.B.4.A, HSF.IF.C.8.A
What is the Answer?
(Solution 1, Solution 2) = ((-1 + √23 i)/2, (-1 - √23 i)/2) Y-intercept = (0,6) Axis of Symmetry: h = -0.5 vertex (h ,k ) = (-0.5 ,5.75) Vertex form = (x + 0.5)2 + 5.75 concave up Factor: (a + 0)(a + 0) Factor: (a + 0)(a + 0)
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 = ax
2 + 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 + kVIDEO 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
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