Solve 24/3627/2x12/7quad2*3/7=frac{15}{45}

$$\frac { 24 } { 36 } \frac { 27 } { 2 x } \frac { 12 } { 7 } \quad 2 \cdot \frac { 3 } { 7 } = \frac { 15 } { 45 }$$

Solve for x

$$x = \frac{1944}{49} = 39\frac{33}{49} \approx 39.673469388$$

Steps Using Factoring

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 1260x, the least common multiple of 36,2x,7,45.

$$35x\times 24\times \frac{27}{2x}\times \frac{12}{7}\times 2\times \frac{3}{7}=28x\times 15$$

Multiply 35 and 24 to get 840.

$$840x\times \frac{27}{2x}\times \frac{12}{7}\times 2\times \frac{3}{7}=28x\times 15$$

Multiply 840 and \frac{12}{7} to get 1440.

$$1440x\times \frac{27}{2x}\times 2\times \frac{3}{7}=28x\times 15$$

Multiply 1440 and 2 to get 2880.

$$2880x\times \frac{27}{2x}\times \frac{3}{7}=28x\times 15$$

Multiply 2880 and \frac{3}{7} to get \frac{8640}{7}.

$$\frac{8640}{7}x\times \frac{27}{2x}=28x\times 15$$

Multiply \frac{8640}{7} times \frac{27}{2x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{8640\times 27}{7\times 2x}x=28x\times 15$$

Cancel out 2 in both numerator and denominator.

$$\frac{27\times 4320}{7x}x=28x\times 15$$

Multiply 28 and 15 to get 420.

$$\frac{27\times 4320}{7x}x=420x$$

Multiply 27 and 4320 to get 116640.

$$\frac{116640}{7x}x=420x$$

Express \frac{116640}{7x}x as a single fraction.

$$\frac{116640x}{7x}=420x$$

Subtract 420x from both sides.

$$\frac{116640x}{7x}-420x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -420x times \frac{7x}{7x}.

$$\frac{116640x}{7x}+\frac{-420x\times 7x}{7x}=0$$

Since \frac{116640x}{7x} and \frac{-420x\times 7x}{7x} have the same denominator, add them by adding their numerators.

$$\frac{116640x-420x\times 7x}{7x}=0$$

Do the multiplications in 116640x-420x\times 7x.

$$\frac{116640x-2940x^{2}}{7x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 7x.

$$116640x-2940x^{2}=0$$

Factor out x.

$$x\left(116640-2940x\right)=0$$

To find equation solutions, solve x=0 and 116640-2940x=0.

$$x=0 x=\frac{1944}{49}$$

Variable x cannot be equal to 0.

$$x=\frac{1944}{49}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 1260x, the least common multiple of 36,2x,7,45.

$$35x\times 24\times \frac{27}{2x}\times \frac{12}{7}\times 2\times \frac{3}{7}=28x\times 15$$

Multiply 35 and 24 to get 840.

$$840x\times \frac{27}{2x}\times \frac{12}{7}\times 2\times \frac{3}{7}=28x\times 15$$

Multiply 840 and \frac{12}{7} to get 1440.

$$1440x\times \frac{27}{2x}\times 2\times \frac{3}{7}=28x\times 15$$

Multiply 1440 and 2 to get 2880.

$$2880x\times \frac{27}{2x}\times \frac{3}{7}=28x\times 15$$

Multiply 2880 and \frac{3}{7} to get \frac{8640}{7}.

$$\frac{8640}{7}x\times \frac{27}{2x}=28x\times 15$$

Multiply \frac{8640}{7} times \frac{27}{2x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{8640\times 27}{7\times 2x}x=28x\times 15$$

Cancel out 2 in both numerator and denominator.

$$\frac{27\times 4320}{7x}x=28x\times 15$$

Multiply 28 and 15 to get 420.

$$\frac{27\times 4320}{7x}x=420x$$

Multiply 27 and 4320 to get 116640.

$$\frac{116640}{7x}x=420x$$

Express \frac{116640}{7x}x as a single fraction.

$$\frac{116640x}{7x}=420x$$

Subtract 420x from both sides.

$$\frac{116640x}{7x}-420x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -420x times \frac{7x}{7x}.

$$\frac{116640x}{7x}+\frac{-420x\times 7x}{7x}=0$$

Since \frac{116640x}{7x} and \frac{-420x\times 7x}{7x} have the same denominator, add them by adding their numerators.

$$\frac{116640x-420x\times 7x}{7x}=0$$

Do the multiplications in 116640x-420x\times 7x.

$$\frac{116640x-2940x^{2}}{7x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 7x.

$$116640x-2940x^{2}=0$$

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.

$$-2940x^{2}+116640x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2940 for a, 116640 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-116640±\sqrt{116640^{2}}}{2\left(-2940\right)}$$

Take the square root of 116640^{2}.

$$x=\frac{-116640±116640}{2\left(-2940\right)}$$

Multiply 2 times -2940.

$$x=\frac{-116640±116640}{-5880}$$

Now solve the equation x=\frac{-116640±116640}{-5880} when ± is plus. Add -116640 to 116640.

$$x=\frac{0}{-5880}$$

Divide 0 by -5880.

$$x=0$$

Now solve the equation x=\frac{-116640±116640}{-5880} when ± is minus. Subtract 116640 from -116640.

$$x=-\frac{233280}{-5880}$$

Reduce the fraction \frac{-233280}{-5880} to lowest terms by extracting and canceling out 120.

$$x=\frac{1944}{49}$$

The equation is now solved.

$$x=0 x=\frac{1944}{49}$$

Variable x cannot be equal to 0.

$$x=\frac{1944}{49}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 1260x, the least common multiple of 36,2x,7,45.

$$35x\times 24\times \frac{27}{2x}\times \frac{12}{7}\times 2\times \frac{3}{7}=28x\times 15$$

Multiply 35 and 24 to get 840.

$$840x\times \frac{27}{2x}\times \frac{12}{7}\times 2\times \frac{3}{7}=28x\times 15$$

Multiply 840 and \frac{12}{7} to get 1440.

$$1440x\times \frac{27}{2x}\times 2\times \frac{3}{7}=28x\times 15$$

Multiply 1440 and 2 to get 2880.

$$2880x\times \frac{27}{2x}\times \frac{3}{7}=28x\times 15$$

Multiply 2880 and \frac{3}{7} to get \frac{8640}{7}.

$$\frac{8640}{7}x\times \frac{27}{2x}=28x\times 15$$

Multiply \frac{8640}{7} times \frac{27}{2x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{8640\times 27}{7\times 2x}x=28x\times 15$$

Cancel out 2 in both numerator and denominator.

$$\frac{27\times 4320}{7x}x=28x\times 15$$

Multiply 28 and 15 to get 420.

$$\frac{27\times 4320}{7x}x=420x$$

Multiply 27 and 4320 to get 116640.

$$\frac{116640}{7x}x=420x$$

Express \frac{116640}{7x}x as a single fraction.

$$\frac{116640x}{7x}=420x$$

Subtract 420x from both sides.

$$\frac{116640x}{7x}-420x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -420x times \frac{7x}{7x}.

$$\frac{116640x}{7x}+\frac{-420x\times 7x}{7x}=0$$

Since \frac{116640x}{7x} and \frac{-420x\times 7x}{7x} have the same denominator, add them by adding their numerators.

$$\frac{116640x-420x\times 7x}{7x}=0$$

Do the multiplications in 116640x-420x\times 7x.

$$\frac{116640x-2940x^{2}}{7x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 7x.

$$116640x-2940x^{2}=0$$

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.

$$-2940x^{2}+116640x=0$$

Divide both sides by -2940.

$$\frac{-2940x^{2}+116640x}{-2940}=\frac{0}{-2940}$$

Dividing by -2940 undoes the multiplication by -2940.

$$x^{2}+\frac{116640}{-2940}x=\frac{0}{-2940}$$

Reduce the fraction \frac{116640}{-2940} to lowest terms by extracting and canceling out 60.

$$x^{2}-\frac{1944}{49}x=\frac{0}{-2940}$$

Divide 0 by -2940.

$$x^{2}-\frac{1944}{49}x=0$$

Divide -\frac{1944}{49}, the coefficient of the x term, by 2 to get -\frac{972}{49}. Then add the square of -\frac{972}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1944}{49}x+\left(-\frac{972}{49}\right)^{2}=\left(-\frac{972}{49}\right)^{2}$$

Square -\frac{972}{49} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1944}{49}x+\frac{944784}{2401}=\frac{944784}{2401}$$

Factor x^{2}-\frac{1944}{49}x+\frac{944784}{2401}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{972}{49}\right)^{2}=\frac{944784}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{972}{49}\right)^{2}}=\sqrt{\frac{944784}{2401}}$$

Simplify.

$$x-\frac{972}{49}=\frac{972}{49} x-\frac{972}{49}=-\frac{972}{49}$$

Add \frac{972}{49} to both sides of the equation.

$$x=\frac{1944}{49} x=0$$

Variable x cannot be equal to 0.

$$x=\frac{1944}{49}$$