\(x^{3} + x^{2} - 16x = 16\)
5.5 Solving cubic equations
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5.5 Solving cubic equations (EMCGX)
Now that we know how to factorise cubic polynomials, it is also easy to solve cubic equations of the form \(a{x}^{3}+b{x}^{2}+cx+d=0\).
Worked example 13: Solving cubic equations
Solve: \(6{x}^{3}-5{x}^{2}-17x+6 = 0\)
Find one factor using the factor theorem
Let \(f(x) = 6{x}^{3}-5{x}^{2}-17x+6\)
Try
\(f\left(1\right)=6{\left(1\right)}^{3}-5{\left(1\right)}^{2}-17\left(1\right)+6=6-5-17+6=-10\)Therefore \(\left(x-1\right)\) is not a factor.
Try
\(f\left(2\right)=6{\left(2\right)}^{3}-5{\left(2\right)}^{2}-17\left(2\right)+6=48-20-34+6=0\)Therefore \(\left(x-2\right)\) is a factor.
Factorise by inspection
\[6{x}^{3}-5{x}^{2}-17x+6=\left(x-2\right)\left(6{x}^{2}+7x-3\right)\]Factorise fully
\[6{x}^{3}-5{x}^{2}-17x+6 = \left(x-2\right) \left(2x+3\right) \left(3x-1\right)\]Solve the equation
\begin{align*} 6{x}^{3}-5{x}^{2}-17x+6 & = 0 \\ \left(x-2\right)\left(2x+3\right)\left(3x-1\right) & = 0 \\ x = 2 \text{ or } x & = \frac{1}{3} \text{ or } x = -\frac{3}{2} \end{align*}Sometimes it is not possible to factorise a quadratic expression using inspection, in which case we use the quadratic formula to fully factorise and solve the cubic equation.
\[x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}\]Worked example 14: Solving cubic equations
Solve for \(x\): \(0 = {x}^{3}-2{x}^{2}-6x+4\)
Use the factor theorem to determine a factor
Let \(f(x)= {x}^{3}-2{x}^{2}-6x+4\)
Try
\(f\left(1\right)={\left(1\right)}^{3}-2{\left(1\right)}^{2}-6\left(1\right)+4=1-2-6+4=-3\)Therefore \(\left(x-1\right)\) is not a factor.
Try
\(f\left(2\right)={\left(2\right)}^{3}-2{\left(2\right)}^{2}-6\left(2\right)+4=8-8-12+4=-8\)Therefore \(\left(x-2\right)\) is not a factor.
\(f\left(-2\right)={\left(-2\right)}^{3}-2{\left(-2\right)}^{2}-6\left(-2\right)+4=-8-8+12+4=0\)
Therefore \(\left(x+2\right)\) is a factor.
Factorise by inspection
\[{x}^{3}-2{x}^{2}-6x+4 = \left(x+2\right) \left({x}^{2}-4x+2\right)\]\({x}^{2}-4x+2\) cannot be factorised any further and we are left with
\(\left(x+2\right)\left({x}^{2}-4x+2\right)=0\)
Solve the equation
\begin{align*} \left(x+2\right)\left({x}^{2}-4x+2\right) & = 0 \\ \left(x+2\right)=0 & \text{ or } \left({x}^{2}-4x+2\right)=0 \end{align*}Apply the quadratic formula for the second bracket
Always write down the formula first and then substitute the values of \(a,b\) and \(c\).
\[a = 1; \qquad b = -4; \qquad c = 2\]
\begin{align*} x & = \frac{-b±\sqrt{{b}^{2}-4ac}}{2a} \\ & = \frac{-\left(-4\right)±\sqrt{{\left(-4\right)}^{2} -4\left(1\right)\left(2\right)}}{2\left(1\right)} \\ & = \frac{4±\sqrt{8}}{2} \\ & = 2±\sqrt{2} \end{align*}Final solutions
\(x=-2\) or \(x=2±\sqrt{2}\)
Solving cubic equations
Solve the following cubic equations:
\(-n^{3} - n^{2} + 22n + 40 = 0\)
\(y(y^{2} + 2y) = 19y + 20\)
\(k^{3} + 9k^{2} + 26k + 24 = 0\)
\(x^{3} + 2x^{2} - 50 = 25x\)
\(-p^{3} + 19p = 30\)
\(6x^{2} - x^{3} = 5x + 12\)
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