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Step-by-step explanation:
Here is your explanation:
((x4) - 3x2) - 18 = 0 Factoring x4-3x2-18
The first term is, x4 its coefficient is 1 .
The middle term is, -3x2 its coefficient is -3 .
The last term, "the constant", is -18
Step-1 : Multiply the coefficient of the first term by the constant 1 • -18 = -18
Step-2 : Find two factors of -18 whose sum equals the coefficient of the middle term, which is -3 .
-18 + 1 = -17 -9 + 2 = -7 -6 + 3 = -3 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -6 and 3
x4 - 6x2 + 3x2 - 18
Step-4 : Add up the first 2 terms, pulling out like factors :
x2 • (x2-6)
Add up the last 2 terms, pulling out common factors :
3 • (x2-6)
Step-5 : Add up the four terms of step 4 :
(x2+3) • (x2-6)
Which is the desired factorization
Find roots (zeroes) of : F(x) = x2+3
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 1 and the Trailing Constant is 3.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3
Let us test ....
P Q P/Q F(P/Q) Divisor -1 1 -1.00 4.00 -3 1 -3.00 12.00 1 1 1.00 4.00 3 1 3.00 12.00
Factoring: x2-6
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 : 6 is not a square !!(x2 + 3) • (x2 - 6) = 0 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.Solve : x2+3 = 0
Subtract 3 from both sides of the equation :
x2 = -3
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
x = ± √ -3
In Math, i is called the imaginary unit. It satisfies i2 =-1. Both i and -i are the square roots of -1
Accordingly, √ -3 =
√ -1• 3 =
√ -1 •√ 3 =
i • √ 3
The equation has no real solutions. It has 2 imaginary, or complex solutions.
x= 0.0000 + 1.7321 i
x= 0.0000 - 1.7321 i
Solve : x2-6 = 0
Add 6 to both sides of the equation :
x2 = 6
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
x = ± √ 6
The equation has two real solutions
These solutions are x = ± √6 = ± 2.4495
Solving x4-3x2-18 = 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
Solve x4-3x2-18 = 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 :
w2-3w-18 = 0
Solving this new equation using the quadratic formula we get two real solutions :
6.0000 or -3.0000
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
x4-3x2-18 = 0
are either :
x =√ 6.000 = 2.44949 or :
x =√ 6.000 = -2.44949 or :
x =√-3.000 = 0.0 + 1.73205 i or :
x =√-3.000 = 0.0 - 1.73205 i
so we found four answers:
x = ± √6 = ± 2.4495
x= 0.0000 - 1.7321 i
x= 0.0000 + 1.7321 i
sorry for taking time to answer
hope its helps
The correct option is A because [tex]x=\pm\sqrt{6}i[/tex] or [tex]x=\pm\sqrt{3}[/tex].
Given:
The given equation is:
[tex]x^4+3x^2-18=0[/tex]
To find:
The real and imaginary solutions of the given equation.
Explanation:
We have,
[tex]x^4+3x^2-18=0[/tex]
Substitute [tex]x^2=t[/tex] in the given equation.
[tex]t^2+3t-18=0[/tex]
Splite the middle term.
[tex]t^2+6t-3t-18=0[/tex]
[tex]t(t+6)-3(t+6)=0[/tex]
[tex](t+6)(t-3)=0[/tex]
Using zero product property, we get
[tex]t+6=0[/tex] or [tex]t-3=0[/tex]
[tex]t=-6[/tex] or [tex]t=3[/tex]
Substituting [tex]t=x^2[/tex], we get
[tex]x^2=-6[/tex] or [tex]x^2=3[/tex]
[tex]x=\pm\sqrt{-6}[/tex] or [tex]x=\pm\sqrt{3}[/tex]
[tex]x=\pm\sqrt{6}i[/tex] or [tex]x=\pm\sqrt{3}[/tex]
Therefore, the correct option is A.
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