Solve 3(x)^(3)-13(x)^(2)+13x-3=0 | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$3 { x }^{ 3 } -13 { x }^{ 2 } +13x-3=0$$

Solve for x

$x=1$
$x=3$
$x=\frac{1}{3}\approx 0.333333333$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $-3$ and $q$ divides the leading coefficient $3$. List all candidates $\frac{p}{q}$.

$$±1,±3,±\frac{1}{3}$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=1$$

By Factor theorem, $x-k$ is a factor of the polynomial for each root $k$. Divide $3x^{3}-13x^{2}+13x-3$ by $x-1$ to get $3x^{2}-10x+3$. Solve the equation where the result equals to $0$.

$$3x^{2}-10x+3=0$$

All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. Substitute $3$ for $a$, $-10$ for $b$, and $3$ for $c$ in the quadratic formula.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 3\times 3}}{2\times 3}$$

Do the calculations.

$$x=\frac{10±8}{6}$$

Solve the equation $3x^{2}-10x+3=0$ when $±$ is plus and when $±$ is minus.

$$x=\frac{1}{3}$$ $$x=3$$

List all found solutions.

$$x=1$$ $$x=\frac{1}{3}$$ $$x=3$$