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 x
2 +2x - 35 = 35 - 35
x
2 +2x - 35 = 0
Set up the a , b , and c values: a = 1 , b = 2 , c = -35
Quadratic Formula
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 = 5
Solution 2 Solution 2 = Numerator 2 Denominator
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
Plug in -b = -2 and a = 1 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
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: LCM of 1 and 4 = 4
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 LCM of 1 and 2 = 2
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 LCM of 1 and 2 = 2
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 (Solution 1, Solution 2) = (5, -7) 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)
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) = (5, -7) 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?
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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