Solve 8x^3+343=0

$$8 { x }^{ 3 } +343 = 0$$

Solve for x (complex solution)

$$x=\frac{-7\sqrt{3}i+7}{4}\approx 1.75-3.031088913i$$

$$x = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

$$x=\frac{7+7\sqrt{3}i}{4}\approx 1.75+3.031088913i$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 343 and q divides the leading coefficient 8. List all candidates \frac{p}{q}.

$$±\frac{343}{8},±\frac{343}{4},±\frac{343}{2},±343,±\frac{49}{8},±\frac{49}{4},±\frac{49}{2},±49,±\frac{7}{8},±\frac{7}{4},±\frac{7}{2},±7,±\frac{1}{8},±\frac{1}{4},±\frac{1}{2},±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=-\frac{7}{2}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 8x^{3}+343 by 2\left(x+\frac{7}{2}\right)=2x+7 to get 4x^{2}-14x+49. Solve the equation where the result equals to 0.

$$4x^{2}-14x+49=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 4 for a, -14 for b, and 49 for c in the quadratic formula.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 4\times 49}}{2\times 4}$$

Do the calculations.

$$x=\frac{14±\sqrt{-588}}{8}$$

Solve the equation 4x^{2}-14x+49=0 when ± is plus and when ± is minus.

$$x=\frac{-7i\sqrt{3}+7}{4} x=\frac{7+7i\sqrt{3}}{4}$$

List all found solutions.

$$x=-\frac{7}{2} x=\frac{-7i\sqrt{3}+7}{4} x=\frac{7+7i\sqrt{3}}{4}$$

Solve for x

$$x = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 343 and q divides the leading coefficient 8. List all candidates \frac{p}{q}.

$$±\frac{343}{8},±\frac{343}{4},±\frac{343}{2},±343,±\frac{49}{8},±\frac{49}{4},±\frac{49}{2},±49,±\frac{7}{8},±\frac{7}{4},±\frac{7}{2},±7,±\frac{1}{8},±\frac{1}{4},±\frac{1}{2},±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=-\frac{7}{2}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 8x^{3}+343 by 2\left(x+\frac{7}{2}\right)=2x+7 to get 4x^{2}-14x+49. Solve the equation where the result equals to 0.

$$4x^{2}-14x+49=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 4 for a, -14 for b, and 49 for c in the quadratic formula.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 4\times 49}}{2\times 4}$$

Do the calculations.

$$x=\frac{14±\sqrt{-588}}{8}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$x\in \emptyset$$

List all found solutions.

$$x=-\frac{7}{2}$$