Find the roots of 4x 2 + x – 5 = 0 by the method of completing the square.
Answers
Answer:
4x2-x-5=0
Two solutions were found :
x = -1
x = 5/4 = 1.250
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 :
(22x2 - x) - 5 = 0
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 4x2-x-5
The first term is, 4x2 its coefficient is 4 .
The middle term is, -x its coefficient is -1 .
The last term, "the constant", is -5
Step-1 : Multiply the coefficient of the first term by the constant 4 • -5 = -20
Step-2 : Find two factors of -20 whose sum equals the coefficient of the middle term, which is -1 .
-20 + 1 = -19
-10 + 2 = -8
-5 + 4 = -1 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -5 and 4
4x2 - 5x + 4x - 5
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (4x-5)
Add up the last 2 terms, pulling out common factors :
1 • (4x-5)
Step-5 : Add up the four terms of step 4 :
(x+1) • (4x-5)
Which is the desired factorization
Equation at the end of step 2 :
(4x - 5) • (x + 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 : 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
Solving a Single Variable Equation :
3.3 Solve : x+1 = 0
Subtract 1 from both sides of the equation :
x = -1
Supplement : Solving Quadratic Equation Directly
Solving 4x2-x-5 = 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
Answer:
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