Solve 2-10x=-2/x-4-46

$$2-10x=- \frac{ 2 }{ x-4 } -46$$

Solve for x

$$x=5$$

$$x = \frac{19}{5} = 3\frac{4}{5} = 3.8$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by x-4.

$$\left(x-4\right)\times 2-10x\left(x-4\right)=-2+\left(x-4\right)\left(-46\right)$$

Use the distributive property to multiply x-4 by 2.

$$2x-8-10x\left(x-4\right)=-2+\left(x-4\right)\left(-46\right)$$

Use the distributive property to multiply -10x by x-4.

$$2x-8-10x^{2}+40x=-2+\left(x-4\right)\left(-46\right)$$

Combine 2x and 40x to get 42x.

$$42x-8-10x^{2}=-2+\left(x-4\right)\left(-46\right)$$

Use the distributive property to multiply x-4 by -46.

$$42x-8-10x^{2}=-2-46x+184$$

Add -2 and 184 to get 182.

$$42x-8-10x^{2}=182-46x$$

Subtract 182 from both sides.

$$42x-8-10x^{2}-182=-46x$$

Subtract 182 from -8 to get -190.

$$42x-190-10x^{2}=-46x$$

Add 46x to both sides.

$$42x-190-10x^{2}+46x=0$$

Combine 42x and 46x to get 88x.

$$88x-190-10x^{2}=0$$

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.

$$-10x^{2}+88x-190=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 88 for b, and -190 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-88±\sqrt{88^{2}-4\left(-10\right)\left(-190\right)}}{2\left(-10\right)}$$

Square 88.

$$x=\frac{-88±\sqrt{7744-4\left(-10\right)\left(-190\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-88±\sqrt{7744+40\left(-190\right)}}{2\left(-10\right)}$$

Multiply 40 times -190.

$$x=\frac{-88±\sqrt{7744-7600}}{2\left(-10\right)}$$

Add 7744 to -7600.

$$x=\frac{-88±\sqrt{144}}{2\left(-10\right)}$$

Take the square root of 144.

$$x=\frac{-88±12}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-88±12}{-20}$$

Now solve the equation x=\frac{-88±12}{-20} when ± is plus. Add -88 to 12.

$$x=-\frac{76}{-20}$$

Reduce the fraction \frac{-76}{-20} to lowest terms by extracting and canceling out 4.

$$x=\frac{19}{5}$$

Now solve the equation x=\frac{-88±12}{-20} when ± is minus. Subtract 12 from -88.

$$x=-\frac{100}{-20}$$

Divide -100 by -20.

$$x=5$$

The equation is now solved.

$$x=\frac{19}{5} x=5$$

Steps for Completing the Square

Variable x cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by x-4.

$$\left(x-4\right)\times 2-10x\left(x-4\right)=-2+\left(x-4\right)\left(-46\right)$$

Use the distributive property to multiply x-4 by 2.

$$2x-8-10x\left(x-4\right)=-2+\left(x-4\right)\left(-46\right)$$

Use the distributive property to multiply -10x by x-4.

$$2x-8-10x^{2}+40x=-2+\left(x-4\right)\left(-46\right)$$

Combine 2x and 40x to get 42x.

$$42x-8-10x^{2}=-2+\left(x-4\right)\left(-46\right)$$

Use the distributive property to multiply x-4 by -46.

$$42x-8-10x^{2}=-2-46x+184$$

Add -2 and 184 to get 182.

$$42x-8-10x^{2}=182-46x$$

Add 46x to both sides.

$$42x-8-10x^{2}+46x=182$$

Combine 42x and 46x to get 88x.

$$88x-8-10x^{2}=182$$

Add 8 to both sides.

$$88x-10x^{2}=182+8$$

Add 182 and 8 to get 190.

$$88x-10x^{2}=190$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-10x^{2}+88x=190$$

Divide both sides by -10.

$$\frac{-10x^{2}+88x}{-10}=\frac{190}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{88}{-10}x=\frac{190}{-10}$$

Reduce the fraction \frac{88}{-10} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{44}{5}x=\frac{190}{-10}$$

Divide 190 by -10.

$$x^{2}-\frac{44}{5}x=-19$$

Divide -\frac{44}{5}, the coefficient of the x term, by 2 to get -\frac{22}{5}. Then add the square of -\frac{22}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{44}{5}x+\left(-\frac{22}{5}\right)^{2}=-19+\left(-\frac{22}{5}\right)^{2}$$

Square -\frac{22}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{44}{5}x+\frac{484}{25}=-19+\frac{484}{25}$$

Add -19 to \frac{484}{25}.

$$x^{2}-\frac{44}{5}x+\frac{484}{25}=\frac{9}{25}$$

Factor x^{2}-\frac{44}{5}x+\frac{484}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{22}{5}\right)^{2}=\frac{9}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{22}{5}\right)^{2}}=\sqrt{\frac{9}{25}}$$

Simplify.

$$x-\frac{22}{5}=\frac{3}{5} x-\frac{22}{5}=-\frac{3}{5}$$

Add \frac{22}{5} to both sides of the equation.

$$x=5 x=\frac{19}{5}$$