p(x)=x^3 -6x^2 +13x -10. find all the solutions to the equation p(x)=0.
i am not allowed to use -b/2A
of course -b/2a does not work. It's not a quadratic.
Furthermore, x = -b/2a gives the location of the vertex of a quadratic, not the roots.
You know that any rational roots must be among ±1,±2,±5,±10
A little synthetic division shows that
p(x) = (x-2)(x^2-4x+5)
now use the quadratic formula to get the complex roots.
To find all the solutions to the equation \(p(x)=0\), where \(p(x)=x^3 -6x^2 +13x -10\), you can use a different method called factoring by grouping.
1. Start by grouping the terms in pairs.
\(p(x) = (x^3 -6x^2) + (13x -10)\)
2. Factor out the greatest common factor from each group.
\(p(x) = x^2(x-6) + 1(13x -10)\)
3. Simplify and combine like terms.
\(p(x) = x^2(x-6) + (13x -10)\)
4. Apply the distributive property to factor out the common factor from both terms.
\(p(x) = x^2(x-6) + 10(13x/10 -1)\)
\(p(x) = x^2(x-6) + 10(13x/10 -1)\)
\(p(x) = x^2(x-6) + 10(13x/10 -10/10)\)
\(p(x) = x^2(x-6) + 10(13x -10)/10\)
\(p(x) = x^2(x-6) + (130x -100)/10\)
5. Combine the fractions by finding a common denominator.
\(p(x) = x^2(x-6) + (130x -100)/10\)
\(p(x) = x^2(x-6) + (13x -10)/10\)
6. Find the common denominator of 10 and add the fractions.
\(p(x) = x^2(x-6) + (13x -10)/10\)
\(p(x) = x^2(x-6) + (13x -10)/10\)
\(p(x) = x^2(x-6) + (13x -10)/10\)
\(p(x) = (10x^2 - 60x + 13x -10)/10\)
7. Combine like terms.
\(p(x) = (10x^2 - 47x -10)/10\)
8. Now that \(p(x)\) is factored, set each factor equal to zero and solve for \(x\).
\(10x^2 - 47x -10 = 0\)
You can use the quadratic formula to solve this equation:
\(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)
Where \(a=10\), \(b=-47\), and \(c=-10\).
Solving for \(x\) using the quadratic formula will give you the solutions to the equation \(p(x)=0\).