Solve 7(x)=x-5x+8/x+3

$$7 ( x ) = \frac { x - 5 x + 8 } { x + 3 }$$

Solve for x

$$x=\frac{\sqrt{849}-25}{14}\approx 0.295543183$$

$$x=\frac{-\sqrt{849}-25}{14}\approx -3.866971755$$

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.

$$7x\left(x+3\right)=x-5x+8$$

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

$$7x^{2}+21x=x-5x+8$$

Combine x and -5x to get -4x.

$$7x^{2}+21x=-4x+8$$

Add 4x to both sides.

$$7x^{2}+21x+4x=8$$

Combine 21x and 4x to get 25x.

$$7x^{2}+25x=8$$

Subtract 8 from both sides.

$$7x^{2}+25x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 25 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\times 7\left(-8\right)}}{2\times 7}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\times 7\left(-8\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-25±\sqrt{625-28\left(-8\right)}}{2\times 7}$$

Multiply -28 times -8.

$$x=\frac{-25±\sqrt{625+224}}{2\times 7}$$

Add 625 to 224.

$$x=\frac{-25±\sqrt{849}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-25±\sqrt{849}}{14}$$

Now solve the equation x=\frac{-25±\sqrt{849}}{14} when ± is plus. Add -25 to \sqrt{849}.

$$x=\frac{\sqrt{849}-25}{14}$$

Now solve the equation x=\frac{-25±\sqrt{849}}{14} when ± is minus. Subtract \sqrt{849} from -25.

$$x=\frac{-\sqrt{849}-25}{14}$$

The equation is now solved.

$$x=\frac{\sqrt{849}-25}{14} x=\frac{-\sqrt{849}-25}{14}$$

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.

$$7x\left(x+3\right)=x-5x+8$$

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

$$7x^{2}+21x=x-5x+8$$

Combine x and -5x to get -4x.

$$7x^{2}+21x=-4x+8$$

Add 4x to both sides.

$$7x^{2}+21x+4x=8$$

Combine 21x and 4x to get 25x.

$$7x^{2}+25x=8$$

Divide both sides by 7.

$$\frac{7x^{2}+25x}{7}=\frac{8}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}+\frac{25}{7}x=\frac{8}{7}$$

Divide \frac{25}{7}, the coefficient of the x term, by 2 to get \frac{25}{14}. Then add the square of \frac{25}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{25}{7}x+\left(\frac{25}{14}\right)^{2}=\frac{8}{7}+\left(\frac{25}{14}\right)^{2}$$

Square \frac{25}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{25}{7}x+\frac{625}{196}=\frac{8}{7}+\frac{625}{196}$$

Add \frac{8}{7} to \frac{625}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{25}{7}x+\frac{625}{196}=\frac{849}{196}$$

Factor x^{2}+\frac{25}{7}x+\frac{625}{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}.

$$\left(x+\frac{25}{14}\right)^{2}=\frac{849}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{25}{14}\right)^{2}}=\sqrt{\frac{849}{196}}$$

Simplify.

$$x+\frac{25}{14}=\frac{\sqrt{849}}{14} x+\frac{25}{14}=-\frac{\sqrt{849}}{14}$$

Subtract \frac{25}{14} from both sides of the equation.

$$x=\frac{\sqrt{849}-25}{14} x=\frac{-\sqrt{849}-25}{14}$$