Solve 6+3-2x/x=4x+7/7-1

$$6 + \frac { 3 - 2 x } { x } = \frac { 4 x + 7 } { 7 } - 1$$

Solve for x

$$x = \frac{\sqrt{70} + 7}{2} \approx 7.683300133$$

$$x=\frac{7-\sqrt{70}}{2}\approx -0.683300133$$

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 7x, the least common multiple of x,7.

$$7x\times 6+7\left(3-2x\right)=x\left(4x+7\right)+7x\left(-1\right)$$

Multiply 7 and 6 to get 42.

$$42x+7\left(3-2x\right)=x\left(4x+7\right)+7x\left(-1\right)$$

Use the distributive property to multiply 7 by 3-2x.

$$42x+21-14x=x\left(4x+7\right)+7x\left(-1\right)$$

Combine 42x and -14x to get 28x.

$$28x+21=x\left(4x+7\right)+7x\left(-1\right)$$

Use the distributive property to multiply x by 4x+7.

$$28x+21=4x^{2}+7x+7x\left(-1\right)$$

Multiply 7 and -1 to get -7.

$$28x+21=4x^{2}+7x-7x$$

Combine 7x and -7x to get 0.

$$28x+21=4x^{2}$$

Subtract 4x^{2} from both sides.

$$28x+21-4x^{2}=0$$

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.

$$-4x^{2}+28x+21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 28 for b, and 21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-4\right)\times 21}}{2\left(-4\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-4\right)\times 21}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-28±\sqrt{784+16\times 21}}{2\left(-4\right)}$$

Multiply 16 times 21.

$$x=\frac{-28±\sqrt{784+336}}{2\left(-4\right)}$$

Add 784 to 336.

$$x=\frac{-28±\sqrt{1120}}{2\left(-4\right)}$$

Take the square root of 1120.

$$x=\frac{-28±4\sqrt{70}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-28±4\sqrt{70}}{-8}$$

Now solve the equation x=\frac{-28±4\sqrt{70}}{-8} when ± is plus. Add -28 to 4\sqrt{70}.

$$x=\frac{4\sqrt{70}-28}{-8}$$

Divide -28+4\sqrt{70} by -8.

$$x=\frac{7-\sqrt{70}}{2}$$

Now solve the equation x=\frac{-28±4\sqrt{70}}{-8} when ± is minus. Subtract 4\sqrt{70} from -28.

$$x=\frac{-4\sqrt{70}-28}{-8}$$

Divide -28-4\sqrt{70} by -8.

$$x=\frac{\sqrt{70}+7}{2}$$

The equation is now solved.

$$x=\frac{7-\sqrt{70}}{2} x=\frac{\sqrt{70}+7}{2}$$

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 7x, the least common multiple of x,7.

$$7x\times 6+7\left(3-2x\right)=x\left(4x+7\right)+7x\left(-1\right)$$

Multiply 7 and 6 to get 42.

$$42x+7\left(3-2x\right)=x\left(4x+7\right)+7x\left(-1\right)$$

Use the distributive property to multiply 7 by 3-2x.

$$42x+21-14x=x\left(4x+7\right)+7x\left(-1\right)$$

Combine 42x and -14x to get 28x.

$$28x+21=x\left(4x+7\right)+7x\left(-1\right)$$

Use the distributive property to multiply x by 4x+7.

$$28x+21=4x^{2}+7x+7x\left(-1\right)$$

Multiply 7 and -1 to get -7.

$$28x+21=4x^{2}+7x-7x$$

Combine 7x and -7x to get 0.

$$28x+21=4x^{2}$$

Subtract 4x^{2} from both sides.

$$28x+21-4x^{2}=0$$

Subtract 21 from both sides. Anything subtracted from zero gives its negation.

$$28x-4x^{2}=-21$$

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.

$$-4x^{2}+28x=-21$$

Divide both sides by -4.

$$\frac{-4x^{2}+28x}{-4}=-\frac{21}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{28}{-4}x=-\frac{21}{-4}$$

Divide 28 by -4.

$$x^{2}-7x=-\frac{21}{-4}$$

Divide -21 by -4.

$$x^{2}-7x=\frac{21}{4}$$

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

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=\frac{21}{4}+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=\frac{21+49}{4}$$

Add \frac{21}{4} to \frac{49}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-7x+\frac{49}{4}=\frac{35}{2}$$

Factor x^{2}-7x+\frac{49}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{35}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{35}{2}}$$

Simplify.

$$x-\frac{7}{2}=\frac{\sqrt{70}}{2} x-\frac{7}{2}=-\frac{\sqrt{70}}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=\frac{\sqrt{70}+7}{2} x=\frac{7-\sqrt{70}}{2}$$