Cubic equation

In <a href="/facts/Algebra/nMdqczjo">algebra</a>, a cubic equation in one variable is an <a href="/facts/Equation/pxFzLAVR">equation</a> of the form

a
        
          x
          
            3
          
        
        +
        b
        
          x
          
            2
          
        
        +
        c
        x
        +
        d
        =
        0
      
    
    {\displaystyle ax^{3}+bx^{2}+cx+d=0}

in which a is not zero.
The solutions of this equation are called <a href="/facts/Root_of_a_function/mlK3dhoT">roots</a> of the <a href="/facts/Cubic_function/t67bqD6c">cubic function</a> defined by the left-hand side of the equation. If all of the <a href="/facts/Coefficients/5Hwq2Wuq">coefficients</a> a, b, c, and d of the cubic equation are <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, then it has at least one real root (this is true for all odd-degree <a href="/facts/Polynomial_function/Lzak8VVx">polynomial functions</a>). All of the roots of the cubic equation can be found by the following means:

<ul><li><a href="/facts/Algebra/nMdqczjo">algebraically</a>: more precisely, they can be expressed by a cubic formula involving the four coefficients, the four basic <a href="/facts/Arithmetic_operations/hvHwkbMV">arithmetic operations</a>, <a href="/facts/Square_root/AfrzfBdQ">square roots</a>, and <a href="/facts/Cube_root/c8klHLFb">cube roots</a>. (This is also true of <a href="/facts/Quadratic_equation/9IMnkJAT">quadratic</a> (second-degree) and <a href="/facts/Quartic_equation/SjjtXkHL">quartic</a> (fourth-degree) equations, but not for higher-degree equations, by the <a href="/facts/Abel%E2%80%93Ruffini_theorem/tEFaRGvW">Abel–Ruffini theorem</a>.)</li>
<li><a href="/facts/Trigonometry/K4hfaC6F">trigonometrically</a></li>
<li><a href="/facts/Numerical_approximation/QT9cYa9z">numerical approximations</a> of the roots can be found using <a href="/facts/Root-finding_algorithm/IxTDkhBq">root-finding algorithms</a> such as <a href="/facts/Newton%27s_method/VlRhI2FL">Newton's method</a>.</li></ul>
The coefficients do not need to be real numbers. Much of what is covered below is valid for coefficients in any <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> with <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> other than 2 and 3. The solutions of the cubic equation do not necessarily belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are irrational (and even non-real) <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>.

Cubic equation open-in-new

Cubic equation