Write the equation in quadratic form and solve it by factoring.


x2(8x + 65)
63
= x

To solve the equation x^2(8x + 65) = 63, we need to convert it into quadratic form and then solve it by factoring. Here's how:

Step 1: Expand the equation
x^2(8x + 65) = 63
8x^3 + 65x^2 = 63

Step 2: Subtract 63 from both sides to move all terms to one side.
8x^3 + 65x^2 - 63 = 0

Step 3: Factor out the highest common factor, which in this case is 1.
x^3 + (65/8)x^2 - 63/8 = 0

Step 4: To factor the quadratic form, we need to find two binomials that multiply together to produce the given polynomial.

Looking at the constant term -63/8, we need to find two numbers whose product is -63/8 and whose sum is (65/8).

Factors of -63/8:
1, -63/8
(-1), 63/8
3, -21/8
(-3), 21/8
7, -9/8
(-7), 9/8

Among these pairs, there is one pair whose sum is (65/8): (-7), 9/8.

Step 5: Now we can write our quadratic equation in factored form:
(x - 7/8)(x + 9/8)(8x + 65/8) = 0

Step 6: Set each factor equal to zero and solve for x:
x - 7/8 = 0 --> x = 7/8
x + 9/8 = 0 --> x = -9/8
8x + 65/8 = 0 --> 8x = -65/8 --> x = -65/64

So, the three solutions to the equation x^2(8x + 65) = 63 are:
x = 7/8, -9/8, and -65/64.