First, we expand the given equation x(x+2)(12-k)=2 to transform it into the standard quadratic form ax² + bx + c = 0. This involves multiplying the terms and moving the constant to the left side of the equation.

First, we expand the given equation x(x+2)(12-k)=2 to transform it into the standard quadratic form ax² + bx + c = 0. This involves multiplying the terms and moving the constant to the left side of the equation.

For a quadratic equation to have equal roots, its discriminant must be zero. The discriminant is given by the formula D = b² - 4ac. We substitute the coefficients a = (12-k), b = 2(12-k), and c = -2 into this formula and set it to zero.

For a quadratic equation to have equal roots, its discriminant must be zero. The discriminant is given by the formula D = b² - 4ac. We substitute the coefficients a = (12-k), b = 2(12-k), and c = -2 into this formula and set it to zero.

Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $(k,\tfrac{k}{2})$ from the line $3x+4y+5=0$ is:

Answer: 15

Step-by-step solution

Step 1: Rearrange the equation into standard quadratic form

First, we expand the given equation $x(x+2)(12-k)=2$ to transform it into the standard quadratic form $ax^2 + bx + c = 0$ . This involves multiplying the terms and moving the constant to the left side of the equation.

Step 2: Apply the condition for equal roots

For a quadratic equation to have equal roots, its discriminant must be zero. The discriminant is given by the formula $D = b^2 - 4ac$ . We substitute the coefficients $a = (12-k)$ , $b = 2(12-k)$ , and $c = -2$ into this formula and set it to zero.

Step 3: Solve for k

We simplify the equation obtained from the discriminant condition. Factoring out $4(12-k)$ allows us to easily find the possible values for $k$ . This gives us two potential values for $k$ : $12$ and $14$ . However, if $k=12$ , the original equation becomes $-2=0$ , which is not possible. Therefore, $k=14$ is the only valid solution.

Step 4: Determine the coordinates of the point

Now that we have found the value of $k=14$ , we can determine the exact coordinates of the given point $(k, \tfrac{k}{2})$ by substituting $k=14$ into the expression.

Step 5: Calculate the distance from the point to the line

Finally, we use the formula for the perpendicular distance from a point $(x_1, y_1)$ to a line $Ax + By + C = 0$ . We substitute the point $(14, 7)$ and the line $3x + 4y + 5 = 0$ into the formula to calculate the distance.

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Let the equation x(x+2)(12−k)=2x(x+2)(12-k)=2x(x+2)(12−k)=2 have equal roots. Then the distance of the point (k,k2)(k,\tfrac{k}{2})(k,2k​) from the line 3x+4y+5=03x+4y+5=03x+4y+5=0 is: