Solve (x^2-x-6)sqrt{x-1}=2x^2-2x-12

$$( x ^ { 2 } - x - 6 ) \sqrt { x - 1 } = 2 x ^ { 2 } - 2 x - 12$$

Solve for x (complex solution)

$$x=5$$

$$x=3$$

$$x=-2$$

Solution Steps

Square both sides of the equation.

$$\left(\left(x^{2}-x-6\right)\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Use the distributive property to multiply x^{2}-x-6 by \sqrt{x-1}.

$$\left(x^{2}\sqrt{x-1}-x\sqrt{x-1}-6\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Square x^{2}\sqrt{x-1}-x\sqrt{x-1}-6\sqrt{x-1}.

$$\left(\sqrt{x-1}\right)^{2}x^{4}-2\left(\sqrt{x-1}\right)^{2}x^{3}-11\left(\sqrt{x-1}\right)^{2}x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$\left(x-1\right)x^{4}-2\left(\sqrt{x-1}\right)^{2}x^{3}-11\left(\sqrt{x-1}\right)^{2}x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Use the distributive property to multiply x-1 by x^{4}.

$$x^{5}-x^{4}-2\left(\sqrt{x-1}\right)^{2}x^{3}-11\left(\sqrt{x-1}\right)^{2}x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$x^{5}-x^{4}-2\left(x-1\right)x^{3}-11\left(\sqrt{x-1}\right)^{2}x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Use the distributive property to multiply -2 by x-1.

$$x^{5}-x^{4}+\left(-2x+2\right)x^{3}-11\left(\sqrt{x-1}\right)^{2}x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

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

$$x^{5}-x^{4}-2x^{4}+2x^{3}-11\left(\sqrt{x-1}\right)^{2}x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Combine -x^{4} and -2x^{4} to get -3x^{4}.

$$x^{5}-3x^{4}+2x^{3}-11\left(\sqrt{x-1}\right)^{2}x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$x^{5}-3x^{4}+2x^{3}-11\left(x-1\right)x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Use the distributive property to multiply -11 by x-1.

$$x^{5}-3x^{4}+2x^{3}+\left(-11x+11\right)x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Use the distributive property to multiply -11x+11 by x^{2}.

$$x^{5}-3x^{4}+2x^{3}-11x^{3}+11x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Combine 2x^{3} and -11x^{3} to get -9x^{3}.

$$x^{5}-3x^{4}-9x^{3}+11x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$x^{5}-3x^{4}-9x^{3}+11x^{2}+12x\left(x-1\right)+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Use the distributive property to multiply 12x by x-1.

$$x^{5}-3x^{4}-9x^{3}+11x^{2}+12x^{2}-12x+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Combine 11x^{2} and 12x^{2} to get 23x^{2}.

$$x^{5}-3x^{4}-9x^{3}+23x^{2}-12x+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$x^{5}-3x^{4}-9x^{3}+23x^{2}-12x+36\left(x-1\right)=\left(2x^{2}-2x-12\right)^{2}$$

Use the distributive property to multiply 36 by x-1.

$$x^{5}-3x^{4}-9x^{3}+23x^{2}-12x+36x-36=\left(2x^{2}-2x-12\right)^{2}$$

Combine -12x and 36x to get 24x.

$$x^{5}-3x^{4}-9x^{3}+23x^{2}+24x-36=\left(2x^{2}-2x-12\right)^{2}$$

Square 2x^{2}-2x-12.

$$x^{5}-3x^{4}-9x^{3}+23x^{2}+24x-36=4x^{4}-8x^{3}-44x^{2}+48x+144$$

Subtract 4x^{4} from both sides.

$$x^{5}-3x^{4}-9x^{3}+23x^{2}+24x-36-4x^{4}=-8x^{3}-44x^{2}+48x+144$$

Combine -3x^{4} and -4x^{4} to get -7x^{4}.

$$x^{5}-7x^{4}-9x^{3}+23x^{2}+24x-36=-8x^{3}-44x^{2}+48x+144$$

Add 8x^{3} to both sides.

$$x^{5}-7x^{4}-9x^{3}+23x^{2}+24x-36+8x^{3}=-44x^{2}+48x+144$$

Combine -9x^{3} and 8x^{3} to get -x^{3}.

$$x^{5}-7x^{4}-x^{3}+23x^{2}+24x-36=-44x^{2}+48x+144$$

Add 44x^{2} to both sides.

$$x^{5}-7x^{4}-x^{3}+23x^{2}+24x-36+44x^{2}=48x+144$$

Combine 23x^{2} and 44x^{2} to get 67x^{2}.

$$x^{5}-7x^{4}-x^{3}+67x^{2}+24x-36=48x+144$$

Subtract 48x from both sides.

$$x^{5}-7x^{4}-x^{3}+67x^{2}+24x-36-48x=144$$

Combine 24x and -48x to get -24x.

$$x^{5}-7x^{4}-x^{3}+67x^{2}-24x-36=144$$

Subtract 144 from both sides.

$$x^{5}-7x^{4}-x^{3}+67x^{2}-24x-36-144=0$$

Subtract 144 from -36 to get -180.

$$x^{5}-7x^{4}-x^{3}+67x^{2}-24x-180=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -180 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±180,±90,±60,±45,±36,±30,±20,±18,±15,±12,±10,±9,±6,±5,±4,±3,±2,±1$$

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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{5}-7x^{4}-x^{3}+67x^{2}-24x-180 by x+2 to get x^{4}-9x^{3}+17x^{2}+33x-90. Solve the equation where the result equals to 0.

$$x^{4}-9x^{3}+17x^{2}+33x-90=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -90 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±90,±45,±30,±18,±15,±10,±9,±6,±5,±3,±2,±1$$

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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-9x^{3}+17x^{2}+33x-90 by x+2 to get x^{3}-11x^{2}+39x-45. Solve the equation where the result equals to 0.

$$x^{3}-11x^{2}+39x-45=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -45 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±45,±15,±9,±5,±3,±1$$

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.

$$x=3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-11x^{2}+39x-45 by x-3 to get x^{2}-8x+15. Solve the equation where the result equals to 0.

$$x^{2}-8x+15=0$$

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 1 for a, -8 for b, and 15 for c in the quadratic formula.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 1\times 15}}{2}$$

Do the calculations.

$$x=\frac{8±2}{2}$$

Solve the equation x^{2}-8x+15=0 when ± is plus and when ± is minus.

$$x=3 x=5$$

List all found solutions.

$$x=-2 x=3 x=5$$

Substitute -2 for x in the equation \left(x^{2}-x-6\right)\sqrt{x-1}=2x^{2}-2x-12.

$$\left(\left(-2\right)^{2}-\left(-2\right)-6\right)\sqrt{-2-1}=2\left(-2\right)^{2}-2\left(-2\right)-12$$

Simplify. The value x=-2 satisfies the equation.

$$0=0$$

Substitute 3 for x in the equation \left(x^{2}-x-6\right)\sqrt{x-1}=2x^{2}-2x-12.

$$\left(3^{2}-3-6\right)\sqrt{3-1}=2\times 3^{2}-2\times 3-12$$

Simplify. The value x=3 satisfies the equation.

$$0=0$$

Substitute 5 for x in the equation \left(x^{2}-x-6\right)\sqrt{x-1}=2x^{2}-2x-12.

$$\left(5^{2}-5-6\right)\sqrt{5-1}=2\times 5^{2}-2\times 5-12$$

Simplify. The value x=5 satisfies the equation.

$$28=28$$

List all solutions of \sqrt{x-1}\left(x^{2}-x-6\right)=2x^{2}-2x-12.

$$x=-2 x=3 x=5$$

Solve for x

$$x=5$$

$$x=3$$

Solution Steps

Square both sides of the equation.

$$\left(\left(x^{2}-x-6\right)\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Use the distributive property to multiply x^{2}-x-6 by \sqrt{x-1}.

$$\left(x^{2}\sqrt{x-1}-x\sqrt{x-1}-6\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Square x^{2}\sqrt{x-1}-x\sqrt{x-1}-6\sqrt{x-1}.

$$\left(\sqrt{x-1}\right)^{2}x^{4}-2\left(\sqrt{x-1}\right)^{2}x^{3}-11\left(\sqrt{x-1}\right)^{2}x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$\left(x-1\right)x^{4}-2\left(\sqrt{x-1}\right)^{2}x^{3}-11\left(\sqrt{x-1}\right)^{2}x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Use the distributive property to multiply x-1 by x^{4}.

$$x^{5}-x^{4}-2\left(\sqrt{x-1}\right)^{2}x^{3}-11\left(\sqrt{x-1}\right)^{2}x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$x^{5}-x^{4}-2\left(x-1\right)x^{3}-11\left(\sqrt{x-1}\right)^{2}x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Use the distributive property to multiply -2 by x-1.

$$x^{5}-x^{4}+\left(-2x+2\right)x^{3}-11\left(\sqrt{x-1}\right)^{2}x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

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

$$x^{5}-x^{4}-2x^{4}+2x^{3}-11\left(\sqrt{x-1}\right)^{2}x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Combine -x^{4} and -2x^{4} to get -3x^{4}.

$$x^{5}-3x^{4}+2x^{3}-11\left(\sqrt{x-1}\right)^{2}x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$x^{5}-3x^{4}+2x^{3}-11\left(x-1\right)x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Use the distributive property to multiply -11 by x-1.

$$x^{5}-3x^{4}+2x^{3}+\left(-11x+11\right)x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Use the distributive property to multiply -11x+11 by x^{2}.

$$x^{5}-3x^{4}+2x^{3}-11x^{3}+11x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Combine 2x^{3} and -11x^{3} to get -9x^{3}.

$$x^{5}-3x^{4}-9x^{3}+11x^{2}+12x\left(\sqrt{x-1}\right)^{2}+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$x^{5}-3x^{4}-9x^{3}+11x^{2}+12x\left(x-1\right)+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Use the distributive property to multiply 12x by x-1.

$$x^{5}-3x^{4}-9x^{3}+11x^{2}+12x^{2}-12x+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Combine 11x^{2} and 12x^{2} to get 23x^{2}.

$$x^{5}-3x^{4}-9x^{3}+23x^{2}-12x+36\left(\sqrt{x-1}\right)^{2}=\left(2x^{2}-2x-12\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$x^{5}-3x^{4}-9x^{3}+23x^{2}-12x+36\left(x-1\right)=\left(2x^{2}-2x-12\right)^{2}$$

Use the distributive property to multiply 36 by x-1.

$$x^{5}-3x^{4}-9x^{3}+23x^{2}-12x+36x-36=\left(2x^{2}-2x-12\right)^{2}$$

Combine -12x and 36x to get 24x.

$$x^{5}-3x^{4}-9x^{3}+23x^{2}+24x-36=\left(2x^{2}-2x-12\right)^{2}$$

Square 2x^{2}-2x-12.

$$x^{5}-3x^{4}-9x^{3}+23x^{2}+24x-36=4x^{4}-8x^{3}-44x^{2}+48x+144$$

Subtract 4x^{4} from both sides.

$$x^{5}-3x^{4}-9x^{3}+23x^{2}+24x-36-4x^{4}=-8x^{3}-44x^{2}+48x+144$$

Combine -3x^{4} and -4x^{4} to get -7x^{4}.

$$x^{5}-7x^{4}-9x^{3}+23x^{2}+24x-36=-8x^{3}-44x^{2}+48x+144$$

Add 8x^{3} to both sides.

$$x^{5}-7x^{4}-9x^{3}+23x^{2}+24x-36+8x^{3}=-44x^{2}+48x+144$$

Combine -9x^{3} and 8x^{3} to get -x^{3}.

$$x^{5}-7x^{4}-x^{3}+23x^{2}+24x-36=-44x^{2}+48x+144$$

Add 44x^{2} to both sides.

$$x^{5}-7x^{4}-x^{3}+23x^{2}+24x-36+44x^{2}=48x+144$$

Combine 23x^{2} and 44x^{2} to get 67x^{2}.

$$x^{5}-7x^{4}-x^{3}+67x^{2}+24x-36=48x+144$$

Subtract 48x from both sides.

$$x^{5}-7x^{4}-x^{3}+67x^{2}+24x-36-48x=144$$

Combine 24x and -48x to get -24x.

$$x^{5}-7x^{4}-x^{3}+67x^{2}-24x-36=144$$

Subtract 144 from both sides.

$$x^{5}-7x^{4}-x^{3}+67x^{2}-24x-36-144=0$$

Subtract 144 from -36 to get -180.

$$x^{5}-7x^{4}-x^{3}+67x^{2}-24x-180=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -180 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±180,±90,±60,±45,±36,±30,±20,±18,±15,±12,±10,±9,±6,±5,±4,±3,±2,±1$$

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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{5}-7x^{4}-x^{3}+67x^{2}-24x-180 by x+2 to get x^{4}-9x^{3}+17x^{2}+33x-90. Solve the equation where the result equals to 0.

$$x^{4}-9x^{3}+17x^{2}+33x-90=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -90 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±90,±45,±30,±18,±15,±10,±9,±6,±5,±3,±2,±1$$

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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-9x^{3}+17x^{2}+33x-90 by x+2 to get x^{3}-11x^{2}+39x-45. Solve the equation where the result equals to 0.

$$x^{3}-11x^{2}+39x-45=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -45 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±45,±15,±9,±5,±3,±1$$

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.

$$x=3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-11x^{2}+39x-45 by x-3 to get x^{2}-8x+15. Solve the equation where the result equals to 0.

$$x^{2}-8x+15=0$$

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 1 for a, -8 for b, and 15 for c in the quadratic formula.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 1\times 15}}{2}$$

Do the calculations.

$$x=\frac{8±2}{2}$$

Solve the equation x^{2}-8x+15=0 when ± is plus and when ± is minus.

$$x=3 x=5$$

List all found solutions.

$$x=-2 x=3 x=5$$

Substitute -2 for x in the equation \left(x^{2}-x-6\right)\sqrt{x-1}=2x^{2}-2x-12. The expression \sqrt{-2-1} is undefined because the radicand cannot be negative.

$$\left(\left(-2\right)^{2}-\left(-2\right)-6\right)\sqrt{-2-1}=2\left(-2\right)^{2}-2\left(-2\right)-12$$

Substitute 3 for x in the equation \left(x^{2}-x-6\right)\sqrt{x-1}=2x^{2}-2x-12.

$$\left(3^{2}-3-6\right)\sqrt{3-1}=2\times 3^{2}-2\times 3-12$$

Simplify. The value x=3 satisfies the equation.

$$0=0$$

Substitute 5 for x in the equation \left(x^{2}-x-6\right)\sqrt{x-1}=2x^{2}-2x-12.

$$\left(5^{2}-5-6\right)\sqrt{5-1}=2\times 5^{2}-2\times 5-12$$

Simplify. The value x=5 satisfies the equation.

$$28=28$$

List all solutions of \sqrt{x-1}\left(x^{2}-x-6\right)=2x^{2}-2x-12.

$$x=3 x=5$$