Solve the cubic equation:

$$x^3+5=0 $$

Quick Answer

Since the discriminant $$\Delta >0$$, the cubic equation has one real root and two conjugate complex roots.

$$ \Delta=6.25$$

$$\begin{cases} x_1=-\sqrt[3]{5} \\ x_2=\dfrac{\sqrt[3]{5}}{2}+\dfrac{\sqrt[3]{5}}{2}\sqrt{3}i \\ x_3=\dfrac{\sqrt[3]{5}}{2}-\dfrac{\sqrt[3]{5}}{2}\sqrt{3}i \end{cases}$$

In decimals,

$$\begin{cases} x_1=-1.7099759466767 \\ x_2=0.5+0.86602540378444i \\ x_3=0.5-0.86602540378444i \end{cases}$$

Detailed Steps on Solution

1. Use the sum of cubes identity

This cubic equation $$x³ + 5=0$$ contains only a cubic term and a constant term. It might be the simplest form of a cubic equation.

Isolate the leading term by moving the constant term to the right hand side.

$$x^3 = -5$$

and then take cubic roots on the both sides, which results in

$$x =\sqrt{5}$$

If the solution is to find only real root, that's all. However, if it is to find all solutions, need more steps since a cubic eqaution normally has three roots.(Some may have duplicate ones)

Factorization by sum of cubes identity

To find complete roots, use the sum of cubes formula to factorize the binomial

$$\Big(x+\sqrt{5}\Big)\Big[(x)^2-x\cdot \sqrt{5}+(\sqrt{5})^2\Big] = 0$$

Now the binomial on the left hand side is transformed to a product of two expressions. If the above equation is to hold true, either of the expressions is equal to zero.

Then we get a linear equation

$$x+\sqrt{5}=0$$

and a quadratic equation.

$$(x)^2-x\cdot \sqrt{5}+(\sqrt{5})^2=0$$

Solve the linear equation $$x+\sqrt{5}=0$$

Subtract the value of the constant term from both sides yields the first solution.

$$x_1=-\sqrt[3]{5}\approx-1.7099759466767$$

Solve the quadratic equation $$(x)^2-x\cdot \sqrt{5}+(\sqrt{5})^2$$

Let $$m = -\sqrt[3]{5}$$, then the equation is simplified as

$$x^2+mx+m^2= 0$$

Using the root identity for a quadratic equation, $$x$$ is found to be the product of $$m$$ and a complex number.

$$\begin{aligned}\\ x&=\dfrac{-b\pm\sqrt{b^2-4ac} }{2a} \\ &=\dfrac{-m\pm m\sqrt{3}i}{2} \\ \end{aligned}$$

Substitute the value of $$m$$, that is, $$m=-\sqrt[3]{5}$$, then we get another two complex roots

$$x_2 =\dfrac{\sqrt[3]{5}}{2}\Big(1+ \sqrt{3}i\Big)$$

$$x_3 =\dfrac{\sqrt[3]{5}}{2}\Big(1- \sqrt{3}i\Big)$$

2. Cubic Root Formula

The equation $$x³ + 5=0$$ has no quadratic term, compared with the general cubic equation. We can use the root formula to calculate the roots direvtly.

$$t^3 +pt+q=0$$

For the equation $$$$, we have $$p=0$$ and $$q = 5$$

Calculate the discriminant

The nature of the roots are determined by the sign of the discriminant.

$$\begin{aligned} \\\Delta&=\dfrac{q^2}{4}+\dfrac{p^3}{27}\\ & =\dfrac{5^2}{4}+0\\ & =\dfrac{25}{4}\\ & =6.25\\ \end{aligned}$$

2.1 Use the root formula directly

If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.

$$t_{1,2,3} =\begin{cases} \sqrt[3]{-\dfrac{q}{2}+\sqrt{\dfrac{q^2}{4}+\dfrac{p^3}{27} } } +\sqrt[3]{-\dfrac{q}{2} -\sqrt{\dfrac{q^2}{4}+\dfrac{p^3}{27} }}& \\ ω\cdotp \sqrt[3]{-\dfrac{q}{2}+\sqrt{\dfrac{q^2}{4}+\dfrac{p^3}{27} } } + \overline{ω} \sqrt[3]{-\dfrac{q}{2} -\sqrt{\dfrac{q^2}{4}+\dfrac{p^3}{27} }}&\ \\ \overline{ω}\cdotp \sqrt[3]{-\dfrac{q}{2}+\sqrt{\dfrac{q^2}{4}+\dfrac{p^3}{27} } } + ω\cdotp \sqrt[3]{-\dfrac{q}{2} -\sqrt{\dfrac{q^2}{4}+\dfrac{p^3}{27} }} \end{cases}$$

in which, $$ ω = \dfrac{-1+i\sqrt{3}}{2} $$ and $$ \overline{ω} =\dfrac{-1-i\sqrt{3}}{2}$$

Substitute the values of $$p, q$$ and $$\Delta$$ which we have calculated. Then,

$$\begin{aligned} \\t_1&=\sqrt[3]{-\dfrac{5}{2}+\sqrt{\dfrac{25}{4}}}+\sqrt[3]{-\dfrac{5}{2}-\sqrt{\dfrac{25}{4}}}\\ & =\sqrt[3]{-\dfrac{5}{2}+\sqrt{\dfrac{5^2}{2^2}}}+\sqrt[3]{-\dfrac{5}{2}-\sqrt{\dfrac{5^2}{2^2}}}\\ & =\sqrt[3]{-\dfrac{5}{2}+\dfrac{5}{2}}+\sqrt[3]{-\dfrac{5}{2}-\dfrac{5}{2}}\\ & =-\sqrt[3]{5}\\ \end{aligned}$$

If we denote

$$R = -\dfrac{q}{2}+\sqrt{\dfrac{q^2}{4}+\dfrac{p^3}{27} }$$

$$\overline{R} = -\dfrac{q}{2} -\sqrt{\dfrac{q^2}{4}+\dfrac{p^3}{27} }$$

then,

$$\sqrt[3]{R} = 0$$, $$\sqrt[3]{\overline{R}} =-\sqrt[3]{5}$$

$$\begin{aligned} \\t_2&= ω\cdotp \sqrt[3]{R}+ \overline{ω} \sqrt[3]{\overline{R} }\\ & =\dfrac{-\sqrt[3]{R}-\sqrt[3]{\overline{R} }}{2} +\dfrac{\sqrt{3}( \sqrt[3]{R} - \sqrt[3]{\overline{R} }) }{2} i\\ & =\dfrac{1}{2}\Big[0-\Big(-\sqrt[3]{5}\Big)\Big]+\dfrac{\sqrt{3}}{2}\Big[0-\Big(-\sqrt[3]{5}\Big)\Big]i\\ & =\dfrac{\sqrt[3]{5}}{2}+\dfrac{\sqrt[3]{5}}{2}\sqrt{3}i\\ \end{aligned}$$

$$\begin{aligned} \\t_3&= \overline{ω}\cdotp \sqrt[3]{R}+ ω\cdotp \sqrt[3]{\overline{R}}\\ & =\dfrac{-\sqrt[3]{R}-\sqrt[3]{\overline{R} }}{2} +\dfrac{\sqrt{3}(- \sqrt[3]{R} + \sqrt[3]{\overline{R} }) }{2}i \\ & =\dfrac{1}{2}\Big[0-\Big(-\sqrt[3]{5}\Big)\Big]-\dfrac{\sqrt{3}}{2}\Big[0-\Big(-\sqrt[3]{5}\Big)\Big]i\\ & =\dfrac{\sqrt[3]{5}}{2}-\dfrac{\sqrt[3]{5}}{2}\sqrt{3}i\\ \end{aligned}$$

3. Summary

In summary, we have tried the method of to explore the solutions of the equation. The cubic equation $$x³ + 5=0$$ is found to have one real root and two complex roots. Exact values and approximations are given below.