Solve 29/14x+16+29/14x+30=0.58/29

$$\frac{ 29 }{ 14x+16 } + \frac{ 29 }{ 14x+30 } = \frac{ 0.58 }{ 29 }$$

Solve for x

$$x=\frac{1427-\sqrt{2102549}}{14}\approx -1.644064032$$

$$x = \frac{\sqrt{2102549} + 1427}{14} \approx 205.50120689$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{15}{7},-\frac{8}{7} since division by zero is not defined. Multiply both sides of the equation by 58\left(7x+8\right)\left(7x+15\right), the least common multiple of 14x+16,14x+30,29.

$$\left(203x+435\right)\times 29+\left(203x+232\right)\times 29=\left(98x^{2}+322x+240\right)\times 0.58$$

Use the distributive property to multiply 203x+435 by 29.

$$5887x+12615+\left(203x+232\right)\times 29=\left(98x^{2}+322x+240\right)\times 0.58$$

Use the distributive property to multiply 203x+232 by 29.

$$5887x+12615+5887x+6728=\left(98x^{2}+322x+240\right)\times 0.58$$

Combine 5887x and 5887x to get 11774x.

$$11774x+12615+6728=\left(98x^{2}+322x+240\right)\times 0.58$$

Add 12615 and 6728 to get 19343.

$$11774x+19343=\left(98x^{2}+322x+240\right)\times 0.58$$

Use the distributive property to multiply 98x^{2}+322x+240 by 0.58.

$$11774x+19343=56.84x^{2}+186.76x+139.2$$

Subtract 56.84x^{2} from both sides.

$$11774x+19343-56.84x^{2}=186.76x+139.2$$

Subtract 186.76x from both sides.

$$11774x+19343-56.84x^{2}-186.76x=139.2$$

Combine 11774x and -186.76x to get 11587.24x.

$$11587.24x+19343-56.84x^{2}=139.2$$

Subtract 139.2 from both sides.

$$11587.24x+19343-56.84x^{2}-139.2=0$$

Subtract 139.2 from 19343 to get 19203.8.

$$11587.24x+19203.8-56.84x^{2}=0$$

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.

$$-56.84x^{2}+11587.24x+19203.8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -56.84 for a, 11587.24 for b, and 19203.8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11587.24±\sqrt{11587.24^{2}-4\left(-56.84\right)\times 19203.8}}{2\left(-56.84\right)}$$

Square 11587.24 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-11587.24±\sqrt{134264130.8176-4\left(-56.84\right)\times 19203.8}}{2\left(-56.84\right)}$$

Multiply -4 times -56.84.

$$x=\frac{-11587.24±\sqrt{134264130.8176+227.36\times 19203.8}}{2\left(-56.84\right)}$$

Multiply 227.36 times 19203.8 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-11587.24±\sqrt{134264130.8176+4366175.968}}{2\left(-56.84\right)}$$

Add 134264130.8176 to 4366175.968 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-11587.24±\sqrt{138630306.7856}}{2\left(-56.84\right)}$$

Take the square root of 138630306.7856.

$$x=\frac{-11587.24±\frac{203\sqrt{2102549}}{25}}{2\left(-56.84\right)}$$

Multiply 2 times -56.84.

$$x=\frac{-11587.24±\frac{203\sqrt{2102549}}{25}}{-113.68}$$

Now solve the equation x=\frac{-11587.24±\frac{203\sqrt{2102549}}{25}}{-113.68} when ± is plus. Add -11587.24 to \frac{203\sqrt{2102549}}{25}.

$$x=\frac{203\sqrt{2102549}-289681}{-113.68\times 25}$$

Divide \frac{-289681+203\sqrt{2102549}}{25} by -113.68 by multiplying \frac{-289681+203\sqrt{2102549}}{25} by the reciprocal of -113.68.

$$x=\frac{1427-\sqrt{2102549}}{14}$$

Now solve the equation x=\frac{-11587.24±\frac{203\sqrt{2102549}}{25}}{-113.68} when ± is minus. Subtract \frac{203\sqrt{2102549}}{25} from -11587.24.

$$x=\frac{-203\sqrt{2102549}-289681}{-113.68\times 25}$$

Divide \frac{-289681-203\sqrt{2102549}}{25} by -113.68 by multiplying \frac{-289681-203\sqrt{2102549}}{25} by the reciprocal of -113.68.

$$x=\frac{\sqrt{2102549}+1427}{14}$$

The equation is now solved.

$$x=\frac{1427-\sqrt{2102549}}{14} x=\frac{\sqrt{2102549}+1427}{14}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{15}{7},-\frac{8}{7} since division by zero is not defined. Multiply both sides of the equation by 58\left(7x+8\right)\left(7x+15\right), the least common multiple of 14x+16,14x+30,29.

$$\left(203x+435\right)\times 29+\left(203x+232\right)\times 29=\left(98x^{2}+322x+240\right)\times 0.58$$

Use the distributive property to multiply 203x+435 by 29.

$$5887x+12615+\left(203x+232\right)\times 29=\left(98x^{2}+322x+240\right)\times 0.58$$

Use the distributive property to multiply 203x+232 by 29.

$$5887x+12615+5887x+6728=\left(98x^{2}+322x+240\right)\times 0.58$$

Combine 5887x and 5887x to get 11774x.

$$11774x+12615+6728=\left(98x^{2}+322x+240\right)\times 0.58$$

Add 12615 and 6728 to get 19343.

$$11774x+19343=\left(98x^{2}+322x+240\right)\times 0.58$$

Use the distributive property to multiply 98x^{2}+322x+240 by 0.58.

$$11774x+19343=56.84x^{2}+186.76x+139.2$$

Subtract 56.84x^{2} from both sides.

$$11774x+19343-56.84x^{2}=186.76x+139.2$$

Subtract 186.76x from both sides.

$$11774x+19343-56.84x^{2}-186.76x=139.2$$

Combine 11774x and -186.76x to get 11587.24x.

$$11587.24x+19343-56.84x^{2}=139.2$$

Subtract 19343 from both sides.

$$11587.24x-56.84x^{2}=139.2-19343$$

Subtract 19343 from 139.2 to get -19203.8.

$$11587.24x-56.84x^{2}=-19203.8$$

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.

$$-56.84x^{2}+11587.24x=-19203.8$$

Divide both sides of the equation by -56.84, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-56.84x^{2}+11587.24x}{-56.84}=-\frac{19203.8}{-56.84}$$

Dividing by -56.84 undoes the multiplication by -56.84.

$$x^{2}+\frac{11587.24}{-56.84}x=-\frac{19203.8}{-56.84}$$

Divide 11587.24 by -56.84 by multiplying 11587.24 by the reciprocal of -56.84.

$$x^{2}-\frac{1427}{7}x=-\frac{19203.8}{-56.84}$$

Divide -19203.8 by -56.84 by multiplying -19203.8 by the reciprocal of -56.84.

$$x^{2}-\frac{1427}{7}x=\frac{2365}{7}$$

Divide -\frac{1427}{7}, the coefficient of the x term, by 2 to get -\frac{1427}{14}. Then add the square of -\frac{1427}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1427}{7}x+\left(-\frac{1427}{14}\right)^{2}=\frac{2365}{7}+\left(-\frac{1427}{14}\right)^{2}$$

Square -\frac{1427}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1427}{7}x+\frac{2036329}{196}=\frac{2365}{7}+\frac{2036329}{196}$$

Add \frac{2365}{7} to \frac{2036329}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1427}{7}x+\frac{2036329}{196}=\frac{2102549}{196}$$

Factor x^{2}-\frac{1427}{7}x+\frac{2036329}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1427}{14}\right)^{2}=\frac{2102549}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1427}{14}\right)^{2}}=\sqrt{\frac{2102549}{196}}$$

Simplify.

$$x-\frac{1427}{14}=\frac{\sqrt{2102549}}{14} x-\frac{1427}{14}=-\frac{\sqrt{2102549}}{14}$$

Add \frac{1427}{14} to both sides of the equation.

$$x=\frac{\sqrt{2102549}+1427}{14} x=\frac{1427-\sqrt{2102549}}{14}$$