Solve xleft(x+4right)+10x=106

$$x \left( x+4 \right) +10x = 106$$

Solve for x (complex solution)

$$x=\sqrt{155}-7\approx 5.449899598$$

$$x=-\left(\sqrt{155}+7\right)\approx -19.449899598$$

Steps Using the Quadratic Formula

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

$$x^{2}+4x+10x=106$$

Combine 4x and 10x to get 14x.

$$x^{2}+14x=106$$

Subtract 106 from both sides.

$$x^{2}+14x-106=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and -106 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-106\right)}}{2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-106\right)}}{2}$$

Multiply -4 times -106.

$$x=\frac{-14±\sqrt{196+424}}{2}$$

Add 196 to 424.

$$x=\frac{-14±\sqrt{620}}{2}$$

Take the square root of 620.

$$x=\frac{-14±2\sqrt{155}}{2}$$

Now solve the equation x=\frac{-14±2\sqrt{155}}{2} when ± is plus. Add -14 to 2\sqrt{155}.

$$x=\frac{2\sqrt{155}-14}{2}$$

Divide -14+2\sqrt{155} by 2.

$$x=\sqrt{155}-7$$

Now solve the equation x=\frac{-14±2\sqrt{155}}{2} when ± is minus. Subtract 2\sqrt{155} from -14.

$$x=\frac{-2\sqrt{155}-14}{2}$$

Divide -14-2\sqrt{155} by 2.

$$x=-\sqrt{155}-7$$

The equation is now solved.

$$x=\sqrt{155}-7 x=-\sqrt{155}-7$$

Steps for Completing the Square

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

$$x^{2}+4x+10x=106$$

Combine 4x and 10x to get 14x.

$$x^{2}+14x=106$$

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

$$x^{2}+14x+7^{2}=106+7^{2}$$

Square 7.

$$x^{2}+14x+49=106+49$$

Add 106 to 49.

$$x^{2}+14x+49=155$$

Factor x^{2}+14x+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}.

$$\left(x+7\right)^{2}=155$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{155}$$

Simplify.

$$x+7=\sqrt{155} x+7=-\sqrt{155}$$

Subtract 7 from both sides of the equation.

$$x=\sqrt{155}-7 x=-\sqrt{155}-7$$

Solve for x

$$x=\sqrt{155}-7\approx 5.449899598$$

$$x=-\sqrt{155}-7\approx -19.449899598$$

Steps Using the Quadratic Formula

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

$$x^{2}+4x+10x=106$$

Combine 4x and 10x to get 14x.

$$x^{2}+14x=106$$

Subtract 106 from both sides.

$$x^{2}+14x-106=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and -106 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-106\right)}}{2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-106\right)}}{2}$$

Multiply -4 times -106.

$$x=\frac{-14±\sqrt{196+424}}{2}$$

Add 196 to 424.

$$x=\frac{-14±\sqrt{620}}{2}$$

Take the square root of 620.

$$x=\frac{-14±2\sqrt{155}}{2}$$

Now solve the equation x=\frac{-14±2\sqrt{155}}{2} when ± is plus. Add -14 to 2\sqrt{155}.

$$x=\frac{2\sqrt{155}-14}{2}$$

Divide -14+2\sqrt{155} by 2.

$$x=\sqrt{155}-7$$

Now solve the equation x=\frac{-14±2\sqrt{155}}{2} when ± is minus. Subtract 2\sqrt{155} from -14.

$$x=\frac{-2\sqrt{155}-14}{2}$$

Divide -14-2\sqrt{155} by 2.

$$x=-\sqrt{155}-7$$

The equation is now solved.

$$x=\sqrt{155}-7 x=-\sqrt{155}-7$$

Steps for Completing the Square

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

$$x^{2}+4x+10x=106$$

Combine 4x and 10x to get 14x.

$$x^{2}+14x=106$$

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

$$x^{2}+14x+7^{2}=106+7^{2}$$

Square 7.

$$x^{2}+14x+49=106+49$$

Add 106 to 49.

$$x^{2}+14x+49=155$$

Factor x^{2}+14x+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}.

$$\left(x+7\right)^{2}=155$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{155}$$

Simplify.

$$x+7=\sqrt{155} x+7=-\sqrt{155}$$

Subtract 7 from both sides of the equation.

$$x=\sqrt{155}-7 x=-\sqrt{155}-7$$