Solve (x-7)(x+4)/5+x+7/7=7

$$\frac { ( x - 7 ) ( x + 4 ) } { 5 } + \frac { x + 7 } { 7 } = 7$$

Solve for x

$$x = \frac{\sqrt{2906} + 8}{7} \approx 8.843903984$$

$$x=\frac{8-\sqrt{2906}}{7}\approx -6.558189698$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 35, the least common multiple of 5,7.

$$7\left(x-7\right)\left(x+4\right)+5\left(x+7\right)=245$$

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

$$\left(7x-49\right)\left(x+4\right)+5\left(x+7\right)=245$$

Use the distributive property to multiply 7x-49 by x+4 and combine like terms.

$$7x^{2}-21x-196+5\left(x+7\right)=245$$

Use the distributive property to multiply 5 by x+7.

$$7x^{2}-21x-196+5x+35=245$$

Combine -21x and 5x to get -16x.

$$7x^{2}-16x-196+35=245$$

Add -196 and 35 to get -161.

$$7x^{2}-16x-161=245$$

Subtract 245 from both sides.

$$7x^{2}-16x-161-245=0$$

Subtract 245 from -161 to get -406.

$$7x^{2}-16x-406=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -16 for b, and -406 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 7\left(-406\right)}}{2\times 7}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 7\left(-406\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-16\right)±\sqrt{256-28\left(-406\right)}}{2\times 7}$$

Multiply -28 times -406.

$$x=\frac{-\left(-16\right)±\sqrt{256+11368}}{2\times 7}$$

Add 256 to 11368.

$$x=\frac{-\left(-16\right)±\sqrt{11624}}{2\times 7}$$

Take the square root of 11624.

$$x=\frac{-\left(-16\right)±2\sqrt{2906}}{2\times 7}$$

The opposite of -16 is 16.

$$x=\frac{16±2\sqrt{2906}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{16±2\sqrt{2906}}{14}$$

Now solve the equation x=\frac{16±2\sqrt{2906}}{14} when ± is plus. Add 16 to 2\sqrt{2906}.

$$x=\frac{2\sqrt{2906}+16}{14}$$

Divide 16+2\sqrt{2906} by 14.

$$x=\frac{\sqrt{2906}+8}{7}$$

Now solve the equation x=\frac{16±2\sqrt{2906}}{14} when ± is minus. Subtract 2\sqrt{2906} from 16.

$$x=\frac{16-2\sqrt{2906}}{14}$$

Divide 16-2\sqrt{2906} by 14.

$$x=\frac{8-\sqrt{2906}}{7}$$

The equation is now solved.

$$x=\frac{\sqrt{2906}+8}{7} x=\frac{8-\sqrt{2906}}{7}$$

Steps for Completing the Square

Multiply both sides of the equation by 35, the least common multiple of 5,7.

$$7\left(x-7\right)\left(x+4\right)+5\left(x+7\right)=245$$

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

$$\left(7x-49\right)\left(x+4\right)+5\left(x+7\right)=245$$

Use the distributive property to multiply 7x-49 by x+4 and combine like terms.

$$7x^{2}-21x-196+5\left(x+7\right)=245$$

Use the distributive property to multiply 5 by x+7.

$$7x^{2}-21x-196+5x+35=245$$

Combine -21x and 5x to get -16x.

$$7x^{2}-16x-196+35=245$$

Add -196 and 35 to get -161.

$$7x^{2}-16x-161=245$$

Add 161 to both sides.

$$7x^{2}-16x=245+161$$

Add 245 and 161 to get 406.

$$7x^{2}-16x=406$$

Divide both sides by 7.

$$\frac{7x^{2}-16x}{7}=\frac{406}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}-\frac{16}{7}x=\frac{406}{7}$$

Divide 406 by 7.

$$x^{2}-\frac{16}{7}x=58$$

Divide -\frac{16}{7}, the coefficient of the x term, by 2 to get -\frac{8}{7}. Then add the square of -\frac{8}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{16}{7}x+\left(-\frac{8}{7}\right)^{2}=58+\left(-\frac{8}{7}\right)^{2}$$

Square -\frac{8}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{16}{7}x+\frac{64}{49}=58+\frac{64}{49}$$

Add 58 to \frac{64}{49}.

$$x^{2}-\frac{16}{7}x+\frac{64}{49}=\frac{2906}{49}$$

Factor x^{2}-\frac{16}{7}x+\frac{64}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{8}{7}\right)^{2}=\frac{2906}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{8}{7}\right)^{2}}=\sqrt{\frac{2906}{49}}$$

Simplify.

$$x-\frac{8}{7}=\frac{\sqrt{2906}}{7} x-\frac{8}{7}=-\frac{\sqrt{2906}}{7}$$

Add \frac{8}{7} to both sides of the equation.

$$x=\frac{\sqrt{2906}+8}{7} x=\frac{8-\sqrt{2906}}{7}$$