Solve 2(x-1)^3=63/4

$$2 ( x - 1 ) ^ { 3 } = 6 \frac { 3 } { 4 }$$

Solve for x (complex solution)

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

$$x=\frac{1+3\sqrt{3}i}{4}\approx 0.25+1.299038106i$$

$$x=\frac{-3\sqrt{3}i+1}{4}\approx 0.25-1.299038106i$$

Solution Steps

Multiply both sides of the equation by 4.

$$8\left(x-1\right)^{3}=6\times 4+3$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

$$8\left(x^{3}-3x^{2}+3x-1\right)=6\times 4+3$$

Use the distributive property to multiply 8 by x^{3}-3x^{2}+3x-1.

$$8x^{3}-24x^{2}+24x-8=6\times 4+3$$

Multiply 6 and 4 to get 24.

$$8x^{3}-24x^{2}+24x-8=24+3$$

Add 24 and 3 to get 27.

$$8x^{3}-24x^{2}+24x-8=27$$

Subtract 27 from both sides.

$$8x^{3}-24x^{2}+24x-8-27=0$$

Subtract 27 from -8 to get -35.

$$8x^{3}-24x^{2}+24x-35=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -35 and q divides the leading coefficient 8. List all candidates \frac{p}{q}.

$$±\frac{35}{8},±\frac{35}{4},±\frac{35}{2},±35,±\frac{7}{8},±\frac{7}{4},±\frac{7}{2},±7,±\frac{5}{8},±\frac{5}{4},±\frac{5}{2},±5,±\frac{1}{8},±\frac{1}{4},±\frac{1}{2},±1$$

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.

$$x=\frac{5}{2}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 8x^{3}-24x^{2}+24x-35 by 2\left(x-\frac{5}{2}\right)=2x-5 to get 4x^{2}-2x+7. Solve the equation where the result equals to 0.

$$4x^{2}-2x+7=0$$

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, -2 for b, and 7 for c in the quadratic formula.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 4\times 7}}{2\times 4}$$

Do the calculations.

$$x=\frac{2±\sqrt{-108}}{8}$$

Solve the equation 4x^{2}-2x+7=0 when ± is plus and when ± is minus.

$$x=\frac{-3i\sqrt{3}+1}{4} x=\frac{1+3i\sqrt{3}}{4}$$

List all found solutions.

$$x=\frac{5}{2} x=\frac{-3i\sqrt{3}+1}{4} x=\frac{1+3i\sqrt{3}}{4}$$

Solve for x

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Solution Steps

Multiply both sides of the equation by 4.

$$8\left(x-1\right)^{3}=6\times 4+3$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

$$8\left(x^{3}-3x^{2}+3x-1\right)=6\times 4+3$$

Use the distributive property to multiply 8 by x^{3}-3x^{2}+3x-1.

$$8x^{3}-24x^{2}+24x-8=6\times 4+3$$

Multiply 6 and 4 to get 24.

$$8x^{3}-24x^{2}+24x-8=24+3$$

Add 24 and 3 to get 27.

$$8x^{3}-24x^{2}+24x-8=27$$

Subtract 27 from both sides.

$$8x^{3}-24x^{2}+24x-8-27=0$$

Subtract 27 from -8 to get -35.

$$8x^{3}-24x^{2}+24x-35=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -35 and q divides the leading coefficient 8. List all candidates \frac{p}{q}.

$$±\frac{35}{8},±\frac{35}{4},±\frac{35}{2},±35,±\frac{7}{8},±\frac{7}{4},±\frac{7}{2},±7,±\frac{5}{8},±\frac{5}{4},±\frac{5}{2},±5,±\frac{1}{8},±\frac{1}{4},±\frac{1}{2},±1$$

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.

$$x=\frac{5}{2}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 8x^{3}-24x^{2}+24x-35 by 2\left(x-\frac{5}{2}\right)=2x-5 to get 4x^{2}-2x+7. Solve the equation where the result equals to 0.

$$4x^{2}-2x+7=0$$

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, -2 for b, and 7 for c in the quadratic formula.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 4\times 7}}{2\times 4}$$

Do the calculations.

$$x=\frac{2±\sqrt{-108}}{8}$$

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.

$$x=\frac{5}{2}$$