Solve x(x+5)+5(x-2)=1

$$x(x+5)+5(x-2)=1$$

Solve for x

$$x=-11$$

$$x=1$$

Steps Using the Quadratic Formula

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

$$x^{2}+5x+5\left(x-2\right)=1$$

Use the distributive property to multiply 5 by x-2.

$$x^{2}+5x+5x-10=1$$

Combine 5x and 5x to get 10x.

$$x^{2}+10x-10=1$$

Subtract 1 from both sides.

$$x^{2}+10x-10-1=0$$

Subtract 1 from -10 to get -11.

$$x^{2}+10x-11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 10 for b, and -11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-11\right)}}{2}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-11\right)}}{2}$$

Multiply -4 times -11.

$$x=\frac{-10±\sqrt{100+44}}{2}$$

Add 100 to 44.

$$x=\frac{-10±\sqrt{144}}{2}$$

Take the square root of 144.

$$x=\frac{-10±12}{2}$$

Now solve the equation x=\frac{-10±12}{2} when ± is plus. Add -10 to 12.

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

Divide 2 by 2.

$$x=1$$

Now solve the equation x=\frac{-10±12}{2} when ± is minus. Subtract 12 from -10.

$$x=-\frac{22}{2}$$

Divide -22 by 2.

$$x=-11$$

The equation is now solved.

$$x=1 x=-11$$

Steps for Completing the Square

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

$$x^{2}+5x+5\left(x-2\right)=1$$

Use the distributive property to multiply 5 by x-2.

$$x^{2}+5x+5x-10=1$$

Combine 5x and 5x to get 10x.

$$x^{2}+10x-10=1$$

Add 10 to both sides.

$$x^{2}+10x=1+10$$

Add 1 and 10 to get 11.

$$x^{2}+10x=11$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=11+5^{2}$$

Square 5.

$$x^{2}+10x+25=11+25$$

Add 11 to 25.

$$x^{2}+10x+25=36$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{36}$$

Simplify.

$$x+5=6 x+5=-6$$

Subtract 5 from both sides of the equation.

$$x=1 x=-11$$