This free math tool finds the roots (zeros) of a given polynomial. The calculator computes exact solutions for quadratic, cubic, and quartic equations. Calculator shows all the work and provides step-by-step on how to find zeros and their multiplicities.
solution
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 3.9149\\[1 em]x_2 &= 0.0851 \end{aligned} $$explanation
The solutions of $ 3x^2-12x+1 = 0 $ are: $ x = 2-\dfrac{\sqrt{ 33 }}{ 3 } ~ \text{and} ~ x = 2+\dfrac{\sqrt{ 33 }}{ 3 }$.
You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this quadratic equation.
The process of finding polynomial roots depends on its degree. The degree is the largest exponent in the polynomial. For example, the degree of polynomial p(x)=8x2+3x-1 is 2. We name polynomials according to their degree. For us, the most interesting ones are: quadratic (degree = 2), Cubic (degree=3) and quartic (degree = 4).
This is the standard form of a quadratic equation is
ax2+bx+c=0
The formula for the roots is
$$ x_1, x_2 = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} $$Example 01: Solve the equation 2x2+3x-14.
In this case we have a=2, b=3, c=-14, so the roots are:
$$ \begin{aligned} x_1, x_2 &= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} \\ x_1, x_2 &= \dfrac{-3 \pm \sqrt{3^2-4 \cdot 2 \cdot (-14)}}{2\cdot2} \\ x_1, x_2 &= \dfrac{-3 \pm \sqrt{9 + 4 \cdot 2 \cdot 14}}{4} \\ x_1, x_2 &= \dfrac{-3 \pm \sqrt{121}}{4} \\ x_1, x_2 &= \dfrac{-3 \pm 11}{4} \\ x_1 &= \dfrac{-3 + 11}{4} = \dfrac{8}{4} = 2 \\ x_2 &= \dfrac{-3 - 11}{4} = \dfrac{-14}{4} = -\dfrac{7}{2} \end{aligned} $$Sometimes, it is much easier not to use a formula for finding the roots of a quadratic equation.
Example 02: Solve the equation 2x2+3x=0.
Because our equation now only has two terms, we can apply factoring. Using factoring, we can reduce an original equation to two simple equations.
$$ \begin{aligned} 2x^2 + 3x &= 0 \\ \color{red}{x} \cdot \left( \color{blue}{2x + 3} \right) &= 0 \\ \color{red}{x = 0} \,\,\, \color{blue}{2x + 3} & \color{blue}{= 0} \\ \color{blue}{2x } & \color{blue}{= -3} \\ \color{blue}{x} &\color{blue}{= -\frac{3}{2}} \end{aligned} $$Example 03: Solve equation 2x2-18=0
This is also a quadratic equation that can be solved without using a quadratic formula.
2x2 - 18 = 0 2x2 = 18 x2 = 9
The last equation actually has two solutions. The first one is obvious
$$ \color{blue}{x_1 = \sqrt{9} = 3} $$and the second one is
$$ \color{blue}{x_2 = -\sqrt{9} = -3 }$$To solve a cubic equation, the best strategy is to guess one of three roots.
Example 04: Solve the equation 2x3-4x2-3x+6=0.
Step 1: Guess one root.
The good candidates for solutions are factors of the last coefficient in the equation. In this example, the last number is -6 so our guesses are:
1, 2, 3, 6, -1, -2, -3 and -6
If we plug in x=2into the equation we get,
$$ 2 \cdot \color{blue}{2}^3 - 4 \cdot \color{blue}{2}^2 - 3 \cdot \color{blue}{2} + 6 = \\\\ 2 \cdot 8 - 4 \cdot 4 - 6 - 6 = 0$$So, x=2 is the root of the equation. Now we have to divide polynomial by x-ROOT.
In this case we divide 2x3-x2-3x+6 by x-2.
(2x3-x2-3x+6)/(x-2) = 2x2-3
Now we use 2x2-3 to find remaining roots
$$ \begin{aligned} 2x^2 - 3 &= 0 \\ 2x^2 &= 3 \\ x^2 &= \frac{3}{2} \\ x_1 & = \sqrt{ \frac{3}{2} } = \frac{\sqrt{6}}{2}\\ x_2 & = -\sqrt{ \frac{3}{2} } = - \frac{\sqrt{6}}{2} \end{aligned} $$To solve cubic equations, we usually use the factoring method.
Example 05: Solve equation 2x3-4x2-3x+6=0.
Notice that a cubic polynomial has four terms, and the most common factoring method for such polynomials is factoring by grouping.
$$ \begin{aligned} 2x^3 - 4x^2 - 3x + 6 &=\color{blue}{2x^3-4x^2} \color{red}{-3x + 6} = \\ &= \color{blue}{2x^2(x-2)} \color{red}{-3(x-2)} = \\ &= (x-2)(2x^2 - 3) \end{aligned} $$Now we can split our equation into two smaller equations, which are much easier to solve. The first one is x-2=0 with a solution x=2, and the second one is 2x2-3=0.
$$ \begin{aligned} 2x^2 - 3 &= 0 \\ x^2 = \frac{3}{2} \\ x_1x_2 = \pm \sqrt{\frac{3}{2}} \end{aligned} $$1. Roots of Polynomials — find roots of linear, quadratic and cubic polynomials.
2. Rational Root Theorem with examples and explanations.