Solve x^2-3/x-1=-4

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

Solve for x (complex solution)

$$x=\sqrt{11}-2\approx 1.31662479$$

$$x=-\left(\sqrt{11}+2\right)\approx -5.31662479$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

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

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

$$x^{2}-3=-4x+4$$

Add 4x to both sides.

$$x^{2}-3+4x=4$$

Subtract 4 from both sides.

$$x^{2}-3+4x-4=0$$

Subtract 4 from -3 to get -7.

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

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-7\right)}}{2}$$

Multiply -4 times -7.

$$x=\frac{-4±\sqrt{16+28}}{2}$$

Add 16 to 28.

$$x=\frac{-4±\sqrt{44}}{2}$$

Take the square root of 44.

$$x=\frac{-4±2\sqrt{11}}{2}$$

Now solve the equation x=\frac{-4±2\sqrt{11}}{2} when ± is plus. Add -4 to 2\sqrt{11}.

$$x=\frac{2\sqrt{11}-4}{2}$$

Divide -4+2\sqrt{11} by 2.

$$x=\sqrt{11}-2$$

Now solve the equation x=\frac{-4±2\sqrt{11}}{2} when ± is minus. Subtract 2\sqrt{11} from -4.

$$x=\frac{-2\sqrt{11}-4}{2}$$

Divide -4-2\sqrt{11} by 2.

$$x=-\sqrt{11}-2$$

The equation is now solved.

$$x=\sqrt{11}-2 x=-\sqrt{11}-2$$

Steps for Completing the Square

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

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

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

$$x^{2}-3=-4x+4$$

Add 4x to both sides.

$$x^{2}-3+4x=4$$

Add 3 to both sides.

$$x^{2}+4x=4+3$$

Add 4 and 3 to get 7.

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

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square 2.

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

Add 7 to 4.

$$x^{2}+4x+4=11$$

Factor x^{2}+4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+2\right)^{2}=11$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{11}$$

Simplify.

$$x+2=\sqrt{11} x+2=-\sqrt{11}$$

Subtract 2 from both sides of the equation.

$$x=\sqrt{11}-2 x=-\sqrt{11}-2$$

Solve for x

$$x=\sqrt{11}-2\approx 1.31662479$$

$$x=-\sqrt{11}-2\approx -5.31662479$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

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

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

$$x^{2}-3=-4x+4$$

Add 4x to both sides.

$$x^{2}-3+4x=4$$

Subtract 4 from both sides.

$$x^{2}-3+4x-4=0$$

Subtract 4 from -3 to get -7.

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

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-7\right)}}{2}$$

Multiply -4 times -7.

$$x=\frac{-4±\sqrt{16+28}}{2}$$

Add 16 to 28.

$$x=\frac{-4±\sqrt{44}}{2}$$

Take the square root of 44.

$$x=\frac{-4±2\sqrt{11}}{2}$$

Now solve the equation x=\frac{-4±2\sqrt{11}}{2} when ± is plus. Add -4 to 2\sqrt{11}.

$$x=\frac{2\sqrt{11}-4}{2}$$

Divide -4+2\sqrt{11} by 2.

$$x=\sqrt{11}-2$$

Now solve the equation x=\frac{-4±2\sqrt{11}}{2} when ± is minus. Subtract 2\sqrt{11} from -4.

$$x=\frac{-2\sqrt{11}-4}{2}$$

Divide -4-2\sqrt{11} by 2.

$$x=-\sqrt{11}-2$$

The equation is now solved.

$$x=\sqrt{11}-2 x=-\sqrt{11}-2$$

Steps for Completing the Square

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

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

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

$$x^{2}-3=-4x+4$$

Add 4x to both sides.

$$x^{2}-3+4x=4$$

Add 3 to both sides.

$$x^{2}+4x=4+3$$

Add 4 and 3 to get 7.

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

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square 2.

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

Add 7 to 4.

$$x^{2}+4x+4=11$$

Factor x^{2}+4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+2\right)^{2}=11$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{11}$$

Simplify.

$$x+2=\sqrt{11} x+2=-\sqrt{11}$$

Subtract 2 from both sides of the equation.

$$x=\sqrt{11}-2 x=-\sqrt{11}-2$$