3x^(2)-2=10 find the zeros
Answers
Answer:
Step-by-step explanation:
3x4-x2-10=0
Four solutions were found :
x= 0.0000 - 1.2910 i
x= 0.0000 + 1.2910 i
x = ± √2 = ± 1.4142
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(3x4 - x2) - 10 = 0
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 3x4-x2-10
The first term is, 3x4 its coefficient is 3 .
The middle term is, -x2 its coefficient is -1 .
The last term, "the constant", is -10
Step-1 : Multiply the coefficient of the first term by the constant 3 • -10 = -30
Step-2 : Find two factors of -30 whose sum equals the coefficient of the middle term, which is -1 .
-30 + 1 = -29
-15 + 2 = -13
-10 + 3 = -7
-6 + 5 = -1 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -6 and 5
3x4 - 6x2 + 5x2 - 10
Step-4 : Add up the first 2 terms, pulling out like factors :
3x2 • (x2-2)
Add up the last 2 terms, pulling out common factors :
5 • (x2-2)
Step-5 : Add up the four terms of step 4 :
(3x2+5) • (x2-2)
Which is the desired factorization
Trying to factor as a Difference of Squares :
2.2 Factoring: x2-2
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 : 2 is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares.
Polynomial Roots Calculator :
2.3 Find roots (zeroes) of : F(x) = 3x2+5
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 3 and the Trailing Constant is 5.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,5
Let us test ....
P Q P / Q F(P/Q) Divisor
-1 1 -1.00 8.00
-1 3 -0.33 5.33
-5 1 -5.00 80.00
-5 3 -1.67 13.33
1 1 1.00 8.00
1 3 0.33 5.33
5 1 5.00 80.00
5 3 1.67 13.33
Polynomial Roots Calculator found no rational roots
Equation at the end of step 2 :
(x2 - 2) • (3x2 + 5) = 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 : x2-2 = 0
Add 2 to both sides of the equation :
x2 = 2
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
x = ± √ 2
The equation has two real solutions
These solutions are x = ± √2 = ± 1.4142
Solving a Single Variable Equation :
3.3 Solve : 3x2+5 = 0
Subtract 5 from both sides of the equation :
3x2 = -5
Divide both sides of the equation by 3:
x2 = -5/3 = -1.667
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
x = ± √ -5/3
In Math, i is called the imaginary unit. It satisfies i2 =-1. Both i and -i are the square roots of -1
Accordingly, √ -5/3 =
√ -1• 5/3 =
√ -1 •√ 5/3 =
i • √ 5/3
The equation has no real solutions. It has 2 imaginary, or complex solutions.
x= 0.0000 + 1.2910 i
x= 0.0000 - 1.2910 i
Supplement : Solving Quadratic Equation Directly
Solving 3x4-x2-10 = 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 3x4-x2-10 = 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 :
3w2-w-10 = 0
Solving this new equation using the quadratic formula we get two real solutions :
2.0000 or -1.6667
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
3x4-x2-10 = 0
are either :
x =√ 2.000 = 1.41421 or :
x =√ 2.000 = -1.41421 or :
x =√-1.667 = 0.0 + 1.29099 i or :
x =√-1.667 = 0.0 - 1.29099 i
Four solutions were found :
x= 0.0000 - 1.2910 i
x= 0.0000 + 1.2910 i
x = ± √2 = ± 1.4142