Solve 3x+7/4x+3=5x+4/2x+6

$$\frac { 3 x + 7 } { 4 x + 3 } = \frac { 5 x + 4 } { 2 x + 6 }$$

Solve for x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$x = -\frac{10}{7} = -1\frac{3}{7} \approx -1.428571429$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,-\frac{3}{4} since division by zero is not defined. Multiply both sides of the equation by 2\left(x+3\right)\left(4x+3\right), the least common multiple of 4x+3,2x+6.

$$\left(2x+6\right)\left(3x+7\right)=\left(4x+3\right)\left(5x+4\right)$$

Use the distributive property to multiply 2x+6 by 3x+7 and combine like terms.

$$6x^{2}+32x+42=\left(4x+3\right)\left(5x+4\right)$$

Use the distributive property to multiply 4x+3 by 5x+4 and combine like terms.

$$6x^{2}+32x+42=20x^{2}+31x+12$$

Subtract 20x^{2} from both sides.

$$6x^{2}+32x+42-20x^{2}=31x+12$$

Combine 6x^{2} and -20x^{2} to get -14x^{2}.

$$-14x^{2}+32x+42=31x+12$$

Subtract 31x from both sides.

$$-14x^{2}+32x+42-31x=12$$

Combine 32x and -31x to get x.

$$-14x^{2}+x+42=12$$

Subtract 12 from both sides.

$$-14x^{2}+x+42-12=0$$

Subtract 12 from 42 to get 30.

$$-14x^{2}+x+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -14 for a, 1 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-14\right)\times 30}}{2\left(-14\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-14\right)\times 30}}{2\left(-14\right)}$$

Multiply -4 times -14.

$$x=\frac{-1±\sqrt{1+56\times 30}}{2\left(-14\right)}$$

Multiply 56 times 30.

$$x=\frac{-1±\sqrt{1+1680}}{2\left(-14\right)}$$

Add 1 to 1680.

$$x=\frac{-1±\sqrt{1681}}{2\left(-14\right)}$$

Take the square root of 1681.

$$x=\frac{-1±41}{2\left(-14\right)}$$

Multiply 2 times -14.

$$x=\frac{-1±41}{-28}$$

Now solve the equation x=\frac{-1±41}{-28} when ± is plus. Add -1 to 41.

$$x=\frac{40}{-28}$$

Reduce the fraction \frac{40}{-28} to lowest terms by extracting and canceling out 4.

$$x=-\frac{10}{7}$$

Now solve the equation x=\frac{-1±41}{-28} when ± is minus. Subtract 41 from -1.

$$x=-\frac{42}{-28}$$

Reduce the fraction \frac{-42}{-28} to lowest terms by extracting and canceling out 14.

$$x=\frac{3}{2}$$

The equation is now solved.

$$x=-\frac{10}{7} x=\frac{3}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,-\frac{3}{4} since division by zero is not defined. Multiply both sides of the equation by 2\left(x+3\right)\left(4x+3\right), the least common multiple of 4x+3,2x+6.

$$\left(2x+6\right)\left(3x+7\right)=\left(4x+3\right)\left(5x+4\right)$$

Use the distributive property to multiply 2x+6 by 3x+7 and combine like terms.

$$6x^{2}+32x+42=\left(4x+3\right)\left(5x+4\right)$$

Use the distributive property to multiply 4x+3 by 5x+4 and combine like terms.

$$6x^{2}+32x+42=20x^{2}+31x+12$$

Subtract 20x^{2} from both sides.

$$6x^{2}+32x+42-20x^{2}=31x+12$$

Combine 6x^{2} and -20x^{2} to get -14x^{2}.

$$-14x^{2}+32x+42=31x+12$$

Subtract 31x from both sides.

$$-14x^{2}+32x+42-31x=12$$

Combine 32x and -31x to get x.

$$-14x^{2}+x+42=12$$

Subtract 42 from both sides.

$$-14x^{2}+x=12-42$$

Subtract 42 from 12 to get -30.

$$-14x^{2}+x=-30$$

Divide both sides by -14.

$$\frac{-14x^{2}+x}{-14}=-\frac{30}{-14}$$

Dividing by -14 undoes the multiplication by -14.

$$x^{2}+\frac{1}{-14}x=-\frac{30}{-14}$$

Divide 1 by -14.

$$x^{2}-\frac{1}{14}x=-\frac{30}{-14}$$

Reduce the fraction \frac{-30}{-14} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{1}{14}x=\frac{15}{7}$$

Divide -\frac{1}{14}, the coefficient of the x term, by 2 to get -\frac{1}{28}. Then add the square of -\frac{1}{28} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{14}x+\left(-\frac{1}{28}\right)^{2}=\frac{15}{7}+\left(-\frac{1}{28}\right)^{2}$$

Square -\frac{1}{28} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{14}x+\frac{1}{784}=\frac{15}{7}+\frac{1}{784}$$

Add \frac{15}{7} to \frac{1}{784} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{14}x+\frac{1}{784}=\frac{1681}{784}$$

Factor x^{2}-\frac{1}{14}x+\frac{1}{784}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{28}\right)^{2}=\frac{1681}{784}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{28}\right)^{2}}=\sqrt{\frac{1681}{784}}$$

Simplify.

$$x-\frac{1}{28}=\frac{41}{28} x-\frac{1}{28}=-\frac{41}{28}$$

Add \frac{1}{28} to both sides of the equation.

$$x=\frac{3}{2} x=-\frac{10}{7}$$