) Find the roots of the quadratic 4x2 – 10x + 6
Answers
Answer:
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Answer:
(1): "x2" was replaced by "x^2".
Step by step solution :
STEP
1
:
Equation at the end of step 1
(22x2 - 10x) - 6 = 0
STEP
2
:
STEP
3
:
Pulling out like terms
3.1 Pull out like factors :
4x2 - 10x - 6 = 2 • (2x2 - 5x - 3)
Trying to factor by splitting the middle term
3.2 Factoring 2x2 - 5x - 3
The first term is, 2x2 its coefficient is 2 .
The middle term is, -5x its coefficient is -5 .
The last term, "the constant", is -3
Step-1 : Multiply the coefficient of the first term by the constant 2 • -3 = -6
Step-2 : Find two factors of -6 whose sum equals the coefficient of the middle term, which is -5 .
-6 + 1 = -5 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -6 and 1
2x2 - 6x + 1x - 3
Step-4 : Add up the first 2 terms, pulling out like factors :
2x • (x-3)
Add up the last 2 terms, pulling out common factors :
1 • (x-3)
Step-5 : Add up the four terms of step 4 :
(2x+1) • (x-3)
Which is the desired factorization
Equation at the end of step
3
:
2 • (x - 3) • (2x + 1) = 0
STEP
4
:
Theory - Roots of a product
4.1 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.
Equations which are never true:
4.2 Solve : 2 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Solving a Single Variable Equation:
4.3 Solve : x-3 = 0
Add 3 to both sides of the equation :
x = 3
Solving a Single Variable Equation:
4.4 Solve : 2x+1 = 0
Subtract 1 from both sides of the equation :
2x = -1
Divide both sides of the equation by 2:
x = -1/2 = -0.500