The product of all the rational roots of the equation (x2−9x+11)2−(x−4)(x−5)=3, is equal to
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Step-by-Step Solution
Step 1: Simplify the equation
First, we expand the product (x-4)(x-5). This gives us x2−5x−4x+20, which simplifies to x2−9x+20. We substitute this back into the original equation.
Step 2: Introduce a substitution
To simplify the equation further, we observe that the term x2−9x appears multiple times. Let y=x2−9x+11. Then x2−9x+20 can be written as (x2−9x+11)+9, which is y+9. Substituting these into the equation gives us a quadratic equation in terms of y.
Step 3: Solve for y
Now we solve the quadratic equation for y. Rearranging the terms, we get y2−y−12=0. Factoring this quadratic equation, we find (y−4)(y+3)=0. This yields two possible values for y: y=4 or y=−3.
Step 4: Substitute back and solve for x
We now substitute the values of y back into our original substitution y=x2−9x+11. This gives us two separate quadratic equations for x. For y=4, we have x2−9x+11=4, which simplifies to x2−9x+7=0. For y=−3, we have x2−9x+11=−3, which simplifies to x2−9x+14=0.
Step 5: Find rational roots
We solve each quadratic equation for x. For x2−9x+7=0, the discriminant is D=(−9)2−4(1)(7)=81−28=53. Since 53 is not a perfect square, the roots are irrational. For x2−9x+14=0, we can factor it as (x−2)(x−7)=0, which gives rational roots x=2 and x=7.
Step 6: Calculate the product of rational roots
The rational roots of the original equation are 2 and 7. The product of these rational roots is 2×7=14.