Enter Quadratic equation/inequality below

Hint Number =

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Answer

↓Steps Explained:↓

Solve, factor, complte the square
find the concavity, vertex, vertex form
axis of symmetry and y-intercept for the quadratic:

n²+3n-18 = 0

Set up the a, b, and c values:

a = 1, b = 3, c = -18

Quadratic Formula

n  =	-b ± √b2 - 4ac
	2a

Calculate -b

-b = -(3)

-b = -3

Calculate the discriminant Δ

Δ = b² - 4ac:

Δ = 3² - 4 x 1 x -18

Δ = 9 - -72

Δ = 81 <--- Discriminant

Since Δ > 0, we expect two real roots.

Take the square root of Δ

√Δ = √(81)

√Δ = 9

-b + Δ:

Numerator 1 = -b + √Δ

Numerator 1 = -3 + 9

Numerator 1 = 6

-b - Δ:

Numerator 2 = -b - √Δ

Numerator 2 = -3 - 9

Numerator 2 = -12

Calculate 2a

Denominator = 2 * a

Denominator = 2 * 1

Denominator = 2

Find Solutions

Solution 1  =	Numerator 1
	Denominator

Solution 1 = 3

Solution 2

Solution 2  =	Numerator 2
	Denominator

Solution 2 = -6

Solution Set

(Solution 1, Solution 2) = (3, -6)

Prove our first answer

(3)² + 3(3) - 18 ? 0

(9) + 918 ? 0

9 + 918 ? 0

0 = 0

Prove our second answer

(-6)² + 3(-6) - 18 ? 0

(36) - 1818 ? 0

36 - 1818 ? 0

0 = 0

(Solution 1, Solution 2) = (3, -6)

Calculate the y-intercept

The y-intercept is the point where x = 0

Set n = 0 in ƒ(n) = n² + 3n - 18=

ƒ(0) = (0)² + 3(0) - 18=

ƒ(0) = 0 + 0 - 18

ƒ(0) = -18  ← Y-Intercept

Y-intercept = (0,-18)

Vertex of a parabola

(h,k) where y = a(x - h)² + k

Use the formula rule.

Our equation coefficients are a = 1, b = 3

The formula rule determines h

h = Axis of Symmetry

h  =	-b
	2a

Plug in -b = -3 and a = 1

h  =	-(3)
	2(1)

h  =	-3
	2

h = -1.5  ← Axis of Symmetry

Calculate k

k = ƒ(h) where h = -1.5

ƒ(h) = n²n18=

ƒ(-1.5) = n²n18=

ƒ(-1.5) = 2.25 - 4.5 - 18

ƒ(-1.5) = -20.25

Our vertex (h,k) = (-1.5,-20.25)

Determine our vertex form:

The vertex form is: a(x - h)² + k

Vertex form = (x + 1.5)² - 20.25

Axis of Symmetry: h = -1.5
vertex (h,k) = (-1.5,-20.25)
Vertex form = (x + 1.5)² - 20.25

Analyze the n² coefficient

Since our n² coefficient of 1 is positive
The parabola formed by the quadratic is concave up

concave up

Add 18 to each side

n² + 3n - 18= + 18 = 0 + 18

n² - 4.5n = 18

Complete the square:

Add an amount to both sides

n² + 3n + ? = 18 + ?

Add (½*middle coefficient)² to each side

Amount to add  =	(1 x 3)2
	(2 x 1)2

Amount to add  =	(3)2
	(2)2

Amount to add  =	9
	4

Amount to add = 9/4

Rewrite our perfect square equation:

n² + 3 + (3/2)² = 18 + (3/2)²

(n + 3/2)² = 18 + 9/4

Simplify Right Side of the Equation:

LCM of 1 and 4 = 4

We multiply 18 by 4 ÷ 1 = 4 and 9 by 4 ÷ 4 = 1

Simplified Fraction  =	18 x 4 + 9 x 1
	4

Simplified Fraction  =	72 + 9
	4

Simplified Fraction  =	81
	4

Our fraction can be reduced down:
Using our GCF of 81 and 4 = 81

Reducing top and bottom by 81 we get
1/0.049382716049383

We set our left side = u

u² = (n + 3/2)²

u has two solutions:

u = +√1/0.049382716049383

u = -√1/0.049382716049383

Replacing u, we get:

n + 3/2 = +1

n + 3/2 = -1

Subtract 3/2 from the both sides

n + 3/2 - 3/2 = +1/1 - 3/2

Simplify right side of the equation

Answer 1 = -1/2

We multiply 1 by 2 ÷ 1 = 2 and -3 by 2 ÷ 2 = 1

Simplified Fraction  =	1 x 2 - 3 x 1
	2

Simplified Fraction  =	2 - 3
	2

Simplified Fraction  =	-1
	2

Subtract 3/2 from the both sides

n + 3/2 - 3/2 = -1/1 - 3/2

Simplify right side of the equation

Answer 2 = -5/2

We multiply -1 by 2 ÷ 1 = 2 and -3 by 2 ÷ 2 = 1

Simplified Fraction  =	-1 x 2 - 3 x 1
	2

Simplified Fraction  =	-2 - 3
	2

Simplified Fraction  =	-5
	2

Build factor pairs:

Since a = 1, find all factor pairs of c = -18
These must have a sum = 3

Factor Pairs of -18	Sum of Factor Pair
1,-18	1 - 18 = -17
2,-9	2 - 9 = -7
3,-6	3 - 6 = -3
6,-3	6 - 3 = 3
9,-2	9 - 2 = 7
18,-1	18 - 1 = 17

We want {6,-3}

Since our a coefficient = 1, we setup our factors

(n + Factor Pair Answer 1)(n + Factor Pair Answer 2)

Factor: (n + 6)(n - 3)

Final Answer

(Solution 1, Solution 2) = (3, -6)
Y-intercept = (0,-18)
Axis of Symmetry: h = -1.5
vertex (h,k) = (-1.5,-20.25)
Vertex form = (x + 1.5)² - 20.25
concave up
Factor: (n + 6)(n - 3)
Factor: (n + 6)(n - 3)

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