How to solve 25x^2-4x^2+25=0
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Answers
Answer:
5
Step-by-step explanation:
Step by step solution :
STEP
1
:
Equation at the end of step 1
((0 - 22x2) + 25x) - 25 = 0
STEP
2
:
STEP
3
:
Pulling out like terms
3.1 Pull out like factors :
-4x2 + 25x - 25 = -1 • (4x2 - 25x + 25)
Trying to factor by splitting the middle term
3.2 Factoring 4x2 - 25x + 25
The first term is, 4x2 its coefficient is 4 .
The middle term is, -25x its coefficient is -25 .
The last term, "the constant", is +25
Step-1 : Multiply the coefficient of the first term by the constant 4 • 25 = 100
Step-2 : Find two factors of 100 whose sum equals the coefficient of the middle term, which is -25 .
-100 + -1 = -101
-50 + -2 = -52
-25 + -4 = -29
-20 + -5 = -25 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -20 and -5
4x2 - 20x - 5x - 25
Step-4 : Add up the first 2 terms, pulling out like factors :
4x • (x-5)
Add up the last 2 terms, pulling out common factors :
5 • (x-5)
Step-5 : Add up the four terms of step 4 :
(4x-5) • (x-5)
Which is the desired factorization
Equation at the end of step
3
:
(5 - x) • (4x - 5) = 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.
Solving a Single Variable Equation:
4.2 Solve : -x+5 = 0
Subtract 5 from both sides of the equation :
-x = -5
Multiply both sides of the equation by (-1) : x = 5
Solving a Single Variable Equation:
4.3 Solve : 4x-5 = 0
Add 5 to both sides of the equation :
4x = 5
Divide both sides of the equation by 4:
x = 5/4 = 1.250
Supplement : Solving Quadratic Equation Directly
Solving 4x2-25x+25 = 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 4x2-25x+25 = 0 by Completing The Square .
Divide both sides of the equation by 4 to have 1 as the coefficient of the first term :
x2-(25/4)x+(25/4) = 0
Subtract 25/4 from both side of the equation :
x2-(25/4)x = -25/4
Now the clever bit: Take the coefficient of x , which is 25/4 , divide by two, giving 25/8 , and finally square it giving 625/64
Add 625/64 to both sides of the equation :
On the right hand side we have :
-25/4 + 625/64 The common denominator of the two fractions is 64 Adding (-400/64)+(625/64) gives 225/64
So adding to both sides we finally get :
x2-(25/4)x+(625/64) = 225/64
Adding 625/64 has completed the left hand side into a perfect square :
x2-(25/4)x+(625/64) =
(x-(25/8)) • (x-(25/8)) =
(x-(25/8))2
Things which are equal to the same thing are also equal to one another. Since
x2-(25/4)x+(625/64) = 225/64 and
x2-(25/4)x+(625/64) = (x-(25/8))2
then, according to the law of transitivity,
(x-(25/8))2 = 225/64
We'll refer to this Equation as Eq. #5.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x-(25/8))2 is
(x-(25/8))2/2 =
(x-(25/8))1 =
x-(25/8)
Now, applying the Square Root Principle to Eq. #5.2.1 we get:
x-(25/8) = √ 225/64
Add 25/8 to both sides to obtain:
x = 25/8 + √ 225/64
Since a square root has two values, one positive and the other negative
x2 - (25/4)x + (25/4) = 0
has two solutions:
x = 25/8 + √ 225/64
or
x = 25/8 - √ 225/64
Note that √ 225/64 can be written as
√ 225 / √ 64 which is 15 / 8
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