The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to

Step 1: Simplify the equation

First, we expand the product $(x-4)(x-5)$. This gives us $x^2 - 5x - 4x + 20$, which simplifies to $x^2 - 9x + 20$. We substitute this back into the original equation.

Step 2: Introduce a substitution

To simplify the equation further, we observe that the term $x^2 - 9x$ appears multiple times. Let $y = x^2 - 9x + 11$. Then $x^2 - 9x + 20$ can be written as $(x^2 - 9x + 11) + 9$, 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 $y^2 - y - 12 = 0$. 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 $y = x^2 - 9x + 11$. This gives us two separate quadratic equations for $x$. For $y=4$, we have $x^2 - 9x + 11 = 4$, which simplifies to $x^2 - 9x + 7 = 0$. For $y=-3$, we have $x^2 - 9x + 11 = -3$, which simplifies to $x^2 - 9x + 14 = 0$.

Step 5: Find rational roots

We solve each quadratic equation for $x$. For $x^2 - 9x + 7 = 0$, the discriminant is $D = (-9)^2 - 4(1)(7) = 81 - 28 = 53$. Since $53$ is not a perfect square, the roots are irrational. For $x^2 - 9x + 14 = 0$, 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 $2 \times 7 = 14$.

More from this chapter

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• Simplify (1 + ²)^5.
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