Solve the Equation, with the Quadratic Formula:
( x + 2)^2 - 2( x + 2) - 15 = 0
Solution:
( x + 2)^2 - 2( x + 2) - 15 = 0 ==> x = 3, or x = - 5
Steps:
( x + 2)^2 - 2( x + 2) - 15 = 0
Expand:
( x + 2)^2 - 2( x + 2) - 15: ==> x^2 + 2x - 15
( x + 2)^2 - 2( x + 2) - 15 ==> ( x + 2)^2: ==> x^2 + 4x + 4
Apply Perfect Square Formula: ( a + b )^2 = a^2 + 2ab + b^2
a = x, b = 2 ==> x^2 + 2x * 2 + 2^2
Simplify:
x^2 + 2x * 2 + 2^2: ==> x^2 + 4x + 4
==> x^2 + 2x * 2 + 2^2
Multiply the Numbers:
2 * 2 = 4 ==> x^2 + 4x + 2^2
2^2 = 4 ==> x^2 + 4x + 4
x^2 + 4x + 4 - 2(x + 2) - 15
Expand:
- 2(x + 2): ==> - 2x - 4 - 2(x + 2)
Apply the distributive Law: a( b + c) ==> ab + ac
a = - 2, b = x, c = 2
==> - 2x + ( - 2) * 2
Apply the Plus (+), and Minus ( - ), Rules:
+ ( - a ) ==> - a ==> - 2x - 2 * 2
Multiply the Numbers: 2 * 2 = 4
x^2 + 4x + 4 - 2x - 4 - 15
Simplify: x^2 + 4x + 4 - 2x - 4 - 15: ==> x^2 + 2x - 15
Group Like Terms:
x^2 + 4x + 4 - 2x - 4 - 15
Add Similar Elements: 4x - 2x ==> 2x
x^2 + 4x + 4 - 2x - 4 - 15
Add / Subtract the Numbers: 4 - 4 - 15 ==> - 15
x^2 + 2x - 15 ==> x^2 + 2x - 15 ==> x^2 + 2x - 15 = 0
Solve With Quadratic Formula / Quadratic Equation Formula:
For a Quadratic Equation of the Form: ax^2 + bx + c = 0,
The Solutions are: x^1, 2 = - b sqrt +/- sqrt b^2 - 4ac / 2a
For a = 1, b = 2, c = - 15, x^1, 2 = - 2 sqrt +/- sqrt 2^2 - 4( - 15) / 2 .1
x = - 2 sqrt +/- sqrt 2^2 - 4( - 15) / 2 .1 ==> 3
= - 2 sqrt +/- sqrt 2^2 - 4( - 15) / 2 .1
Apply Rule: - ( - a ) ==> a
= - 2 sqrt +/- sqrt 2^2 - 4( - 15) / 2 .1
sqrt 2^2 + 4 * 1 * 15 = 64
sqrt 2^2 + 4 * 1 * 15
2^2 = 4
sqrt 4 + 4 * 1 * 15
Multiply Numbers: 4 * 1 * 15 = 60
= sqrt 4 + 60
Add Numbers: 4 + 60 = 64
sqrt 64 ==> - 2 + sqrt 64 / 2 * 1
Multiply the Numbers: 2 * 1 = 2
- 2 + sqrt 64 / 2
sqrt 64 = 8
sqrt 64
Factor the Number: 64 = 8^2
==> sqrt 8^2
Apply Radical Rule: n sqrt a^n = a
sqrt 8^2 = 8 ==> 8 ==> - 2 + 8/2
Add / Subtract the Numbers: - 2 + 8 = 6 ==> 6/2
Divid the Numbers: 6/2 = 3 ==> 3
x = - 2 - sqrt 2^2 - 4 * 1 ( - 15) / 2 * 1 ==> - 5
- 2 - sqrt 2^2 - 4 * 1 ( - 15) / 2 * 1
Apply Rule: - ( - a ) = a
- 2 - sqrt 2^2 - 4 * 1 ( - 15) / 2 * 1
sqrt 2^2 + 4 * 1 ( - 15) = sqrt 64
sqrt 2^2 + 4 * 1 ( - 15)
2^2 = 4
sqrt 4 + 4 * 1 * 15
Multiply the Numbers: 4 * 1 * 15 = 60
sqrt 4 + 60
Add the Numbers: 4 + 60 = 64 ==> sqrt 64
= - 2 - sqrt 64 / 2 * 1
Multiply the Numbers: 2 * 1 = 2
= - 2 - sqrt 64 / 2
sqrt 64 = 8 ==> sqrt 64
Factor the Number: 64 = 8^2 ==> sqrt 8^2
Apply Radical Rule: n sqrt a^n = a
sqrt 8^2 = 8 ==> 8 ==> - 2 - 8 / 2
Subtract: - 2 - 8 = - 10 ==> - 10 / 2
Apply the Fraction Rule: - a / b = - a / b ==> - 10 / 2
Divide Numbers: 10 / 2 = 5 ==> - 5
Therefore, The Final Solution / Answers, to your Quadratic Equations to:
(x + 2)^2 - 2(x+2) - 15 =0 ==> x = 3, and x = - 5,
Check the upload for Graph of: ( x + 2)^2 - 2( x + 2) - 15 = 0,
Answers: (x = 3), (x = - 5).
Hope that helps!!!!! : )