In acute triangle ABC, segment AD is an altitude, the measure of angle ADB=2x^2+40, BD=9x-3, and DC=6x+15. Find BC.

To find the length of segment BC, we need to use the given information about segment AD to create an equation involving BC.

In a triangle, the sum of all the angles is equal to 180 degrees. Since AD is an altitude, angle ADB is a right angle (90 degrees). Hence, the other angle of the triangle, angle BDC, can be found by subtracting the measure of angle ADB from 180 degrees.

Measure of angle BDC = 180 - (2x^2 + 40) = 180 - 2x^2 - 40 = 140 - 2x^2

Since AD is an altitude, we can create similar triangles by looking at triangles ADB and BDC.

Using the similar triangles, we can set up a proportion:

(Length of segment BD) / (Length of segment DC) = (Length of segment AB) / (Length of segment BC)

Substituting the given values, we have:

(9x - 3) / (6x + 15) = AB / BC

Cross-multiplying, we get:

(9x - 3) * BC = AB * (6x + 15)

Expanding the equation, we have:

9xBC - 3xBC = 6xABx + 15xAB

Combining like terms, we get:

6xBC = 15xAB + 9xBC

Simplifying further, we have:

6xBC - 9xBC = 15xAB

-3xBC = 15xAB

Dividing both sides by -3x, we get:

BC = -5xAB

Now, we need to find the value of AB in terms of x. To do this, we can use the Pythagorean theorem in triangle ADB.

According to the Pythagorean theorem:

(Length of segment AB)^2 + (Length of segment BD)^2 = (Length of segment AD)^2

Substituting the given values, we have:

(AB)^2 + (9x - 3)^2 = (AD)^2

Simplifying the equation, we have:

AB^2 + 81x^2 - 54x + 9 = AD^2

Since AD is an altitude, AD^2 is equal to (CD)^2, which we can find using the given values:

(6x + 15)^2 = (36x^2 + 180x + 225)

Substituting this value into our equation, we have:

AB^2 + 81x^2 - 54x + 9 = 36x^2 + 180x + 225

Simplifying further, we have:

AB^2 - 36x^2 + 126x - 216 = 0

Now, we have a quadratic equation in terms of AB. We need to solve for AB to find its value.

We can use the quadratic formula to solve this equation:

AB = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = 126, and c = -216.

Plugging these values into the quadratic formula, we have:

AB = (-126 ± √(126^2 - 4(1)(-216))) / (2(1))

Simplifying further, we have:

AB = (-126 ± √(15876 + 864)) / 2

AB = (-126 ± √16740) / 2

Now, we have two potential values for AB. However, one of the values will not work because AB cannot be negative. Therefore, we will only consider the positive square root:

AB = (-126 + √16740) / 2

Now that we have the length of segment AB in terms of x, we can substitute this value back into the equation for BC:

BC = -5xAB

BC = -5((-126 + √16740) / 2)

Simplifying further, we have:

BC = -5(-63 + √16740)

BC = 5(63 - √16740)

Hence, the length of segment BC is 5(63 - √16740).