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# Learn Cubic Equation - Definition, Example, Formula

 Cubic Equation Definition: A cubic equation is a polynomial equation of the third degree. The general form is ax3+bx2+cx+d=0, where a ≠ 0. Cubic Equation: ax3 + bx2 + cx + d = 0, where a = coefficient of x3 b = coefficient of x2 c = coefficient of x and d = constant. Formulae: x1 = -term1 + r13 * cos(q3 / 3) x2 = -term1 + r13 * cos(q3 + (2 * ∏) / 3) x3 = -term1 + r13 * cos(q3 + (4 * ∏) / 3) term1 and r13 formula: q = (3c - b2) / 9 r = (-27d + b(9c - 2b2)) / 54 discriminant(Δ) = q3 + r2 r13 = 2 * √ (q) if(discriminant<0) term1 = (b/3.0) else s = r + √ (discriminant) t = r - √ (discriminant) term1 = √ (3.0) * ((-t + s) / 2) Example: Calculate the roots(x1, x2, x3) of the cubic equation, x 3 - 4x2 - 9x + 36 = 0 Step 1: From the above equation, the value of a = 1, b = - 4, c = - 9 and d = 36. Step 2: Find values of q and r q = (3c - b2) / 9 q = ((3*-9) - (-4)2) / 9 = -4.77778 Step 3: Find value of discriminate, denoted generally as delta(Δ) discriminant(Δ) = q3 + r2 Δ = (-4.77778)3 + (-9.62963)2 = -16.3333 Here the discriminant value is less than 0 Step 4: Find term1 and r13 If,Δ<0, term1 = (b/3.0) = -4 / 3 = -1.33333 term1 = -1.33333 r13 = 2 * √(q) where, q = -q = 4.77778 r13 = 2 * √ 4.77778 = 4.371626 Step 5: To Substitute term1 and r13 values to Cubic formula x1 = -term1 + r13 * cos(q3 / 3) x1 = 1.33333 + 4.371626 x cos(4.777783 / 3) = 4 x2 = -term1 + r13 * cos(q3 + (2 * ∏) / 3) x2 = 1.33333 + 4.371626 x cos(4.777783 + (2 * ∏)/ 3) = -3 x3 = -term1 + r13 * cos(q3 + (4 * ∏) / 3) x3 = 1.33333 + 4.371626 x cos(4.777783 + (4 * ∏)/ 3) = -3 Step 6: We get the roots, x1 = 4, x2 = -3 and x3 = -3. This is an example for real roots in the cubic equation.

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