find the difference of the roots of quadratic equation 25x² - 10x + 1 =0
Pls solve it
Answers
Answer:
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(52x2 - 10x) + 1 > 0
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 25x2-10x+1
The first term is, 25x2 its coefficient is 25 .
The middle term is, -10x its coefficient is -10 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 25 • 1 = 25
Step-2 : Find two factors of 25 whose sum equals the coefficient of the middle term, which is -10 .
-25 + -1 = -26
-5 + -5 = -10 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -5 and -5
25x2 - 5x - 5x - 1
Step-4 : Add up the first 2 terms, pulling out like factors :
5x • (5x-1)
Add up the last 2 terms, pulling out common factors :
1 • (5x-1)
Step-5 : Add up the four terms of step 4 :
(5x-1) • (5x-1)
Which is the desired factorization
Multiplying Exponential Expressions :
2.2 Multiply (5x-1) by (5x-1)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (5x-1) and the exponents are :
1 , as (5x-1) is the same number as (5x-1)1
and 1 , as (5x-1) is the same number as (5x-1)1
The product is therefore, (5x-1)(1+1) = (5x-1)2
Equation at the end of step 2 :
(5x - 1)2 > 0
Step 3 :
3.1 Divide both sides by 5
(x-(1/5))2 > 0
Solve Basic Inequality :
3.2 Add 1/5 to both sides
(x)2 > 1/5
Inequality Plot :
3.3 Inequality plot for
5.000 x - 1.000 > 0
Supplement : Solving Quadratic Equation Directly
Solving 25x2-10x+1 = 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
Parabola, Finding the Vertex :
4.1 Find the Vertex of y = 25x2-10x+1
Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 25 , is positive (greater than zero).
Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.
Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.
For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is 0.2000
Plugging into the parabola formula 0.2000 for x we can calculate the y -coordinate :
y = 25.0 * 0.20 * 0.20 - 10.0 * 0.20 + 1.0
or y = 0.000
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = 25x2-10x+1
Vertex at {x,y} = { 0.20, 0.00}
x-Intercept (Root) :
One Root at {x,y}={ 0.20, 0.00}
Note that the root coincides with
the Vertex and the Axis of Symmetry
coinsides with the line x = 0
Step-by-step explanation:
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