Question

5 (x-6)^3 - 13 (2+x )^3 +32= -8x^2 (x+21)

Answer 1

5(х³-18х²+108х-216)-13(8+12х+6х²+х³)+32=-8х³-168х²

5х³-90х²+540х-1080-104-156х-78х²-13х³+32=-8х³-168х²

-8х³-168х²+384х-1152=-8х³-168х²

384х-1152=0

384х=1152

х=3

Answer 2

To solve the equation $5(x-6)^3 - 13(2+x)^3 + 32 = -8x^2(x+21)$, we will first expand the cubic terms and then simplify the equation step by step. Here's how you can do it:

Step 1: Expand the cubic terms. $(x-6)^3 = x^3 - 3x^2 \cdot 6 + 3x \cdot 6^2 - 6^3 = x^3 - 18x^2 + 108x - 216$ $(2+x)^3 = (2+x)(2+x)^2 = (2+x)(4+4x+x^2) = 8 + 12x + 6x^2 + x^3$

Step 2: Substitute the expanded cubic terms back into the equation: $5(x-6)^3 - 13(2+x)^3 + 32 = 5(x^3 - 18x^2 + 108x - 216) - 13(8 + 12x + 6x^2 + x^3) + 32$

Step 3: Simplify the equation further: $5x^3 - 90x^2 + 540x - 1080 - 104 - 156x - 78x^2 - 13x^3 + 32 = -8x^2(x + 21)$

Step 4: Combine like terms: $5x^3 - 13x^3 - 90x^2 - 78x^2 - 8x^2x + 540x - 156x - 1080 - 104 + 32 = -8x^3 - 168x^2$

Step 5: Arrange the equation in standard form (descending powers of x): $-8x^3 - 168x^2 - 8x^2x + 540x - 156x - 90x^2 - 104 - 1080 + 32 = 0$

Step 6: Combine like terms again: $-8x^3 - 258x^2 - 8x^2x + 384x - 1152 = 0$

Step 7: Factor out common terms: $-8x^2(x + 21) - 8x(x + 21) + 384(x + 21) = 0$

Step 8: Factor out the common factor (x + 21): $(-8x^2 - 8x + 384)(x + 21) = 0$

Step 9: Set each factor equal to zero and solve for x: $x + 21 = 0$ or $-8x^2 - 8x + 384 = 0$

For the first factor: $x = -21$

For the second factor, let's solve the quadratic equation: $-8x^2 - 8x + 384 = 0$

Divide the whole equation by -8 to simplify: $x^2 + x - 48 = 0$

Now, we can solve this quadratic equation by factoring or using the quadratic formula. Factoring gives: $(x + 8)(x - 6) = 0$

So, the solutions are: $x = -21$, $x = -8$, and $x = 6$

These are the three solutions to the original equation.