x4-10x2+9=0
Four solutions were found :
x = 3
x = -3
x = 1
x = -1
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((x4) - (2•5x2)) + 9 = 0
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring x4-10x2+9
The first term is, x4 its coefficient is 1 .
The middle term is, -10x2 its coefficient is -10 .
The last term, "the constant", is +9
Step-1 : Multiply the coefficient of the first term by the constant 1 • 9 = 9
Step-2 : Find two factors of 9 whose sum equals the coefficient of the middle term, which is -10 .
-9 + -1 = -10 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -9 and -1
x4 - 9x2 - 1x2 - 9
Step-4 : Add up the first 2 terms, pulling out like factors :
x2 • (x2-9)
Add up the last 2 terms, pulling out common factors :
1 • (x2-9)
Step-5 : Add up the four terms of step 4 :
(x2-1) • (x2-9)
Which is the desired factorization
Trying to factor as a Difference of Squares :
2.2 Factoring: x2-1
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 : 1 is the square of 1
Check : x2 is the square of x1
Factorization is : (x + 1) • (x - 1)
Trying to factor as a Difference of Squares :
2.3 Factoring: x2 - 9
Check : 9 is the square of 3
Check : x2 is the square of x1
Factorization is : (x + 3) • (x - 3)
Equation at the end of step 2 :
(x + 1) • (x - 1) • (x + 3) • (x - 3) = 0
Step 3 :
Theory - Roots of a product :
3.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Solving a Single Variable Equation :
3.2 Solve : x+1 = 0
Subtract 1 from both sides of the equation :
x = -1
Solving a Single Variable Equation :
3.3 Solve : x-1 = 0
Add 1 to both sides of the equation :
x = 1
Solving a Single Variable Equation :
3.4 Solve : x+3 = 0
Subtract 3 from both sides of the equation :
x = -3
Solving a Single Variable Equation :
3.5 Solve : x-3 = 0
Add 3 to both sides of the equation :
x = 3
Supplement : Solving Quadratic Equation Directly
Solving x4-10x2+9 = 0 directly
Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula
Solving a Single Variable Equation :
Equations which are reducible to quadratic :
4.1 Solve x4-10x2+9 = 0
This equation is reducible to quadratic. What this means is that using a new variable, we can rewrite this equation as a quadratic equation Using w , such that w = x2 transforms the equation into :
w2-10w+9 = 0
Solving this new equation using the quadratic formula we get two real solutions :
9.0000 or 1.0000
Now that we know the value(s) of w , we can calculate x since x is √ w
Doing just this we discover that the solutions of
x4-10x2+9 = 0
are either :
x =√ 9.000 = 3.00000 or :
x =√ 9.000 = -3.00000 or :
x =√ 1.000 = 1.00000 or :
x =√ 1.000 = -1.00000
Four solutions were found :
x = 3
x = -3
x = 1
x = -1
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