find the roots of quadratic equation 9x2-9x+2=0
Answers
Step-by-step explanation:
9x2-9x+2=0
Two solutions were found :
x = 1/3 = 0.333
x = 2/3 = 0.667
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "x2" was replaced by "x^2".
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(32x2 - 9x) + 2 = 0
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 9x2-9x+2
The first term is, 9x2 its coefficient is 9 .
The middle term is, -9x its coefficient is -9 .
The last term, "the constant", is +2
Step-1 : Multiply the coefficient of the first term by the constant 9 • 2 = 18
Step-2 : Find two factors of 18 whose sum equals the coefficient of the middle term, which is -9 .
-18 + -1 = -19
-9 + -2 = -11
-6 + -3 = -9 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -6 and -3
9x2 - 6x - 3x - 2
Step-4 : Add up the first 2 terms, pulling out like factors :
3x • (3x-2)
Add up the last 2 terms, pulling out common factors :
1 • (3x-2)
Step-5 : Add up the four terms of step 4 :
(3x-1) • (3x-2)
Which is the desired factorization
Equation at the end of step 2 :
(3x - 2) • (3x - 1) = 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 : 3x-2 = 0
Add 2 to both sides of the equation :
3x = 2
Divide both sides of the equation by 3:
x = 2/3 = 0.667
Solving a Single Variable Equation :
3.3 Solve : 3x-1 = 0
Add 1 to both sides of the equation :
3x = 1
Divide both sides of the equation by 3:
x = 1/3 = 0.333
Supplement : Solving Quadratic Equation Directly
Solving 9x2-9x+2 = 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
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