Solve 1050/x-10+1050/x=42

$$\frac { 1050 } { x - 10 } + \frac { 1050 } { x } = 42$$

Solve for x

$$x=5\sqrt{26}+30\approx 55.495097568$$

$$x=30-5\sqrt{26}\approx 4.504902432$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,10 since division by zero is not defined. Multiply both sides of the equation by x\left(x-10\right), the least common multiple of x-10,x.

$$x\times 1050+\left(x-10\right)\times 1050=42x\left(x-10\right)$$

Use the distributive property to multiply x-10 by 1050.

$$x\times 1050+1050x-10500=42x\left(x-10\right)$$

Combine x\times 1050 and 1050x to get 2100x.

$$2100x-10500=42x\left(x-10\right)$$

Use the distributive property to multiply 42x by x-10.

$$2100x-10500=42x^{2}-420x$$

Subtract 42x^{2} from both sides.

$$2100x-10500-42x^{2}=-420x$$

Add 420x to both sides.

$$2100x-10500-42x^{2}+420x=0$$

Combine 2100x and 420x to get 2520x.

$$2520x-10500-42x^{2}=0$$

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.

$$-42x^{2}+2520x-10500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -42 for a, 2520 for b, and -10500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2520±\sqrt{2520^{2}-4\left(-42\right)\left(-10500\right)}}{2\left(-42\right)}$$

Square 2520.

$$x=\frac{-2520±\sqrt{6350400-4\left(-42\right)\left(-10500\right)}}{2\left(-42\right)}$$

Multiply -4 times -42.

$$x=\frac{-2520±\sqrt{6350400+168\left(-10500\right)}}{2\left(-42\right)}$$

Multiply 168 times -10500.

$$x=\frac{-2520±\sqrt{6350400-1764000}}{2\left(-42\right)}$$

Add 6350400 to -1764000.

$$x=\frac{-2520±\sqrt{4586400}}{2\left(-42\right)}$$

Take the square root of 4586400.

$$x=\frac{-2520±420\sqrt{26}}{2\left(-42\right)}$$

Multiply 2 times -42.

$$x=\frac{-2520±420\sqrt{26}}{-84}$$

Now solve the equation x=\frac{-2520±420\sqrt{26}}{-84} when ± is plus. Add -2520 to 420\sqrt{26}.

$$x=\frac{420\sqrt{26}-2520}{-84}$$

Divide -2520+420\sqrt{26} by -84.

$$x=30-5\sqrt{26}$$

Now solve the equation x=\frac{-2520±420\sqrt{26}}{-84} when ± is minus. Subtract 420\sqrt{26} from -2520.

$$x=\frac{-420\sqrt{26}-2520}{-84}$$

Divide -2520-420\sqrt{26} by -84.

$$x=5\sqrt{26}+30$$

The equation is now solved.

$$x=30-5\sqrt{26} x=5\sqrt{26}+30$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,10 since division by zero is not defined. Multiply both sides of the equation by x\left(x-10\right), the least common multiple of x-10,x.

$$x\times 1050+\left(x-10\right)\times 1050=42x\left(x-10\right)$$

Use the distributive property to multiply x-10 by 1050.

$$x\times 1050+1050x-10500=42x\left(x-10\right)$$

Combine x\times 1050 and 1050x to get 2100x.

$$2100x-10500=42x\left(x-10\right)$$

Use the distributive property to multiply 42x by x-10.

$$2100x-10500=42x^{2}-420x$$

Subtract 42x^{2} from both sides.

$$2100x-10500-42x^{2}=-420x$$

Add 420x to both sides.

$$2100x-10500-42x^{2}+420x=0$$

Combine 2100x and 420x to get 2520x.

$$2520x-10500-42x^{2}=0$$

Add 10500 to both sides. Anything plus zero gives itself.

$$2520x-42x^{2}=10500$$

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.

$$-42x^{2}+2520x=10500$$

Divide both sides by -42.

$$\frac{-42x^{2}+2520x}{-42}=\frac{10500}{-42}$$

Dividing by -42 undoes the multiplication by -42.

$$x^{2}+\frac{2520}{-42}x=\frac{10500}{-42}$$

Divide 2520 by -42.

$$x^{2}-60x=\frac{10500}{-42}$$

Divide 10500 by -42.

$$x^{2}-60x=-250$$

Divide -60, the coefficient of the x term, by 2 to get -30. Then add the square of -30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-60x+\left(-30\right)^{2}=-250+\left(-30\right)^{2}$$

Square -30.

$$x^{2}-60x+900=-250+900$$

Add -250 to 900.

$$x^{2}-60x+900=650$$

Factor x^{2}-60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-30\right)^{2}=650$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-30\right)^{2}}=\sqrt{650}$$

Simplify.

$$x-30=5\sqrt{26} x-30=-5\sqrt{26}$$

Add 30 to both sides of the equation.

$$x=5\sqrt{26}+30 x=30-5\sqrt{26}$$