Solution - Nonlinear equations
Step by Step Solution
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
3*x^2+2*(x^2-20)-(16-2*x^2)=0
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((3•(x2))+(2•((x2)-20)))-(16-2x2) = 0
Step 2 :
Trying to factor as a Difference of Squares :
2.1 Factoring: x2-20
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 20 is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares.
Equation at the end of step 2 :
((3•(x2))+2•(x2-20))-(16-2x2) = 0Step 3 :
Equation at the end of step 3 :
(3x2 + 2 • (x2 - 20)) - (16 - 2x2) = 0
Step 4 :
Step 5 :
Pulling out like terms :
5.1 Pull out like factors :
7x2 - 56 = 7 • (x2 - 8)
Trying to factor as a Difference of Squares :
5.2 Factoring: x2 - 8
Check : 8 is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares.
Equation at the end of step 5 :
7 • (x2 - 8) = 0
Step 6 :
Equations which are never true :
6.1 Solve : 7 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Solving a Single Variable Equation :
6.2 Solve : x2-8 = 0
Add 8 to both sides of the equation :
x2 = 8
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
x = ± √ 8
Can √ 8 be simplified ?
Yes! The prime factorization of 8 is
2•2•2
To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).
√ 8 = √ 2•2•2 =
± 2 • √ 2
The equation has two real solutions
These solutions are x = 2 • ± √2 = ± 2.8284
Two solutions were found :
x = 2 • ± √2 = ± 2.8284How did we do?
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