Solve 2x-7/x+3=7

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

Solve for x

$$x = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

$$x=4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

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

Use the distributive property to multiply 2x by x+3.

$$2x^{2}+6x-7=7\left(x+3\right)$$

Use the distributive property to multiply 7 by x+3.

$$2x^{2}+6x-7=7x+21$$

Subtract 7x from both sides.

$$2x^{2}+6x-7-7x=21$$

Combine 6x and -7x to get -x.

$$2x^{2}-x-7=21$$

Subtract 21 from both sides.

$$2x^{2}-x-7-21=0$$

Subtract 21 from -7 to get -28.

$$2x^{2}-x-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -1 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 2\left(-28\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-1\right)±\sqrt{1-8\left(-28\right)}}{2\times 2}$$

Multiply -8 times -28.

$$x=\frac{-\left(-1\right)±\sqrt{1+224}}{2\times 2}$$

Add 1 to 224.

$$x=\frac{-\left(-1\right)±\sqrt{225}}{2\times 2}$$

Take the square root of 225.

$$x=\frac{-\left(-1\right)±15}{2\times 2}$$

The opposite of -1 is 1.

$$x=\frac{1±15}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{1±15}{4}$$

Now solve the equation x=\frac{1±15}{4} when ± is plus. Add 1 to 15.

$$x=\frac{16}{4}$$

Divide 16 by 4.

$$x=4$$

Now solve the equation x=\frac{1±15}{4} when ± is minus. Subtract 15 from 1.

$$x=-\frac{14}{4}$$

Reduce the fraction \frac{-14}{4} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

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

Use the distributive property to multiply 2x by x+3.

$$2x^{2}+6x-7=7\left(x+3\right)$$

Use the distributive property to multiply 7 by x+3.

$$2x^{2}+6x-7=7x+21$$

Subtract 7x from both sides.

$$2x^{2}+6x-7-7x=21$$

Combine 6x and -7x to get -x.

$$2x^{2}-x-7=21$$

Add 7 to both sides.

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

Add 21 and 7 to get 28.

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

Divide both sides by 2.

$$\frac{2x^{2}-x}{2}=\frac{28}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{1}{2}x=\frac{28}{2}$$

Divide 28 by 2.

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

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=14+\left(-\frac{1}{4}\right)^{2}$$

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=14+\frac{1}{16}$$

Add 14 to \frac{1}{16}.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{225}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{225}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{225}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{15}{4} x-\frac{1}{4}=-\frac{15}{4}$$

Add \frac{1}{4} to both sides of the equation.

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