Let ( x(square)+ 3x +1) over (2 x(square) -5x+2) = k ----------be y
1) Express y in the form of a x(square) + bx +c =0
2) If -3/4 is a root of y, find the value of k
I don't really understand why the answer is like that. Can you explain?
(x^2+3x+1)/(2x^2-5x+2) = k
x^2+3x+1 = 2kx^2 - 10kx+2k
(1-2k)x^2 + (3+10k)x + (1-2k) = 0
since -3/4 is a root,
(1-2k)(9/16) + (3+10k)(-3/4) + (1-2k) = 0
-1/16 (170k+11) = 0
k = -11/170
( x ^ 2 + 3 x + 1 ) / ( 2 x ^ 2 - 5 x + 2 ) = k Multiply both sides by 2 x ^ 2 - 5 x + 2
x ^ 2 + 3 x + 1 = k ( 2 x ^ 2 - 5 x + 2 )
x ^ 2 + 3 x + 1 = 2 k x ^ 2 - 5 k x + 2 k
x ^ 2 + 3 x + 1 - 2 k x ^ 2 + 5 k x - 2 k = 0
( 1 - 2 k ) x ^ 2 + ( 3 + 5 k ) x + 1 - 2 k = 0
y = ( 1 - 2 k ) x ^ 2 + ( 3 + 5 k ) x + 1 - 2 k = 0
Now we try to find k
We now :
for x = - 3 / 4 , y = 0
0 = ( 1 - 2 k ) x ^ 2 + ( 3 + 5 k ) x + 1 - 2 k
( 1 - 2 k ) x ^ 2 + ( 3 + 5 k ) x + 1 = 0
( 1 - 2 k ) * ( - 3 / 4 ) ^ 2 + ( 3 + 5 k ) * ( - 3 / 4 ) + 1 - 2 k = 0
( 1 - 2 k ) * 9 / 16 + ( 3 + 5 k ) * ( - 3 / 4 ) + 1 - 2 k = 0 Multiply both sides by 16
( 1 - 2 k ) * 9 * 16 / 16 + ( 3 + 5 k ) * ( - 3 / 4 ) * 16 + 1 * 16 - 2 k * 16 = 0
( 1 - 2 k ) * 9 + ( 3 + 5 k ) * ( - 3 / 4 ) * 4 * 4 + 16 - 32 k = 0
( 1 - 2 k ) * 9 + ( 3 + 5 k ) * ( - 3 ) * 4 + 16 - 32 k = 0
( 1 - 2 k ) * 9 + ( 3 + 5 k ) * ( - 12 ) + 16 - 32 k = 0
1 * 9 - 2 k * 9 + 3 * ( - 12 ) + 5 k * ( - 12 ) + 16 - 32 k = 0
9 - 18 k - 36 - 60 k + 16 - 32 k = 0
- 110 k - 11 = 0
- 110 k = 11 Divide both sides by - 100
k = 11 / - 110
k = - 1 / 10
Correction:
- 110 k = 11 Divide both sides by - 110
k = 11 / - 110
k = - 1 / 10
To solve this problem, we are given the equation (x^2 + 3x + 1)/(2x^2 - 5x + 2) = k, and we need to express it in the form of ax^2 + bx + c = 0.
Step 1: Multiply both sides of the equation by (2x^2 - 5x + 2) to eliminate the denominator:
(x^2 + 3x + 1) = k(2x^2 - 5x + 2)
Step 2: Expand both sides of the equation:
x^2 + 3x + 1 = 2kx^2 - 5kx + 2k
Step 3: Rearrange the equation so that all terms are on one side:
(1 - 2k)x^2 + (3 + 10k)x + (1 - 2k) = 0
Now, we are asked to find the value of k if -3/4 is a root of the equation:
Step 4: Substitute x = -3/4 into the equation:
(1 - 2k)(9/16) + (3 + 10k)(-3/4) + (1 - 2k) = 0
Step 5: Simplify the equation:
-9/16 + (3 + 10k)(-3/4) + 1 - 2k = 0
-9/16 - 9/4 - 27/4 + 1 - 2k = 0
-1/16 (170k + 11) = 0
Step 6: Set the expression equal to zero to solve for k:
170k + 11 = 0
170k = -11
k = -11/170
And that is the value of k when -3/4 is a root of the equation.