Factor 12x^2 - 176x + 480 as completely as possible, and find its roots.

Step-by-step solution

1. 1

   Pull out the greatest common factor. The coefficients $12$, $176$ and $480$ are all divisible by $4$ (and by no larger common factor, since $12=4\cdot3$ and $3\nmid176$):

   $12x^2-176x+480=4\left(3x^2-44x+120\right)$

   Check each: $4\cdot3=12$ ✓, $4\cdot44=176$ ✓, $4\cdot120=480$ ✓.

2. 2

   Attempt the AC method on the inner trinomial. Factoring $3x^2-44x+120$ over the integers needs two numbers with product $3\times120=360$ and sum $-44$. The factor pairs of $360$ are $(1,360),(2,180),(3,120),(4,90),(5,72),(6,60),(8,45),(9,40),(10,36),(12,30),(15,24),(18,20)$, giving sums $361,182,123,94,77,66,53,49,46,42,39,38$ — and negating both members of a pair only negates the sum. None equals $44$, so no integer factorisation exists.

3. 3

   Confirm with the discriminant. For $3x^2-44x+120$:

   $\Delta=(-44)^2-4(3)(120)=1936-1440=496$

   and $496=16\times31$ is not a perfect square ($22^2=484$, $23^2=529$). A non-square discriminant is exactly the criterion for "does not factor over the rationals".

4. 4

   Reject a factorisation that circulates for this problem. The answer $4(x-6)(3x-20)$ is sometimes given, but expanding it yields

   $4(x-6)(3x-20)=4\left(3x^2-38x+120\right)=12x^2-152x+480$

   The middle coefficient is $-152$, not $-176$. Substituting $x=1$ settles it: the original gives $12-176+480=316$, while $4(1-6)(3-20)=4(-5)(-17)=340$ — not equal.

5. 5

   Find the exact roots with the quadratic formula. Solving $3x^2-44x+120=0$:

   $x=\frac{44\pm\sqrt{496}}{6}=\frac{44\pm4\sqrt{31}}{6}=\frac{22\pm2\sqrt{31}}{3}$

   where $\sqrt{496}=\sqrt{16\cdot31}=4\sqrt{31}$. Numerically, with $\sqrt{31}=5.56776$:

   $x_1=\frac{22+11.13553}{3}=11.04518,\qquad x_2=\frac{22-11.13553}{3}=3.62149$

6. 6

   State the complete factorisation and verify. Over the reals,

   $12x^2-176x+480=12\left(x-\frac{22+2\sqrt{31}}{3}\right)\left(x-\frac{22-2\sqrt{31}}{3}\right)$

   while over the integers the best possible is $4\left(3x^2-44x+120\right)$. Both roots evaluate the original expression to within $10^{-9}$ of zero ✓, and the GCF step was checked at $40$ integer values ✓.