Solve 154=(11+x)*(7+x)

$$154 = ( 11 + x ) \cdot ( 7 + x )$$

Solve for x (complex solution)

$$x=\sqrt{158}-9\approx 3.56980509$$

$$x=-\left(\sqrt{158}+9\right)\approx -21.56980509$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 11+x by 7+x and combine like terms.

$$154=77+18x+x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$77+18x+x^{2}=154$$

Subtract 154 from both sides.

$$77+18x+x^{2}-154=0$$

Subtract 154 from 77 to get -77.

$$-77+18x+x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+18x-77=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 18 for b, and -77 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-77\right)}}{2}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-77\right)}}{2}$$

Multiply -4 times -77.

$$x=\frac{-18±\sqrt{324+308}}{2}$$

Add 324 to 308.

$$x=\frac{-18±\sqrt{632}}{2}$$

Take the square root of 632.

$$x=\frac{-18±2\sqrt{158}}{2}$$

Now solve the equation x=\frac{-18±2\sqrt{158}}{2} when ± is plus. Add -18 to 2\sqrt{158}.

$$x=\frac{2\sqrt{158}-18}{2}$$

Divide -18+2\sqrt{158} by 2.

$$x=\sqrt{158}-9$$

Now solve the equation x=\frac{-18±2\sqrt{158}}{2} when ± is minus. Subtract 2\sqrt{158} from -18.

$$x=\frac{-2\sqrt{158}-18}{2}$$

Divide -18-2\sqrt{158} by 2.

$$x=-\sqrt{158}-9$$

The equation is now solved.

$$x=\sqrt{158}-9 x=-\sqrt{158}-9$$

Steps for Completing the Square

Use the distributive property to multiply 11+x by 7+x and combine like terms.

$$154=77+18x+x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$77+18x+x^{2}=154$$

Subtract 77 from both sides.

$$18x+x^{2}=154-77$$

Subtract 77 from 154 to get 77.

$$18x+x^{2}=77$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+18x=77$$

Divide 18, the coefficient of the x term, by 2 to get 9. Then add the square of 9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+18x+9^{2}=77+9^{2}$$

Square 9.

$$x^{2}+18x+81=77+81$$

Add 77 to 81.

$$x^{2}+18x+81=158$$

Factor x^{2}+18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+9\right)^{2}=158$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{158}$$

Simplify.

$$x+9=\sqrt{158} x+9=-\sqrt{158}$$

Subtract 9 from both sides of the equation.

$$x=\sqrt{158}-9 x=-\sqrt{158}-9$$

Solve for x

$$x=\sqrt{158}-9\approx 3.56980509$$

$$x=-\sqrt{158}-9\approx -21.56980509$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 11+x by 7+x and combine like terms.

$$154=77+18x+x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$77+18x+x^{2}=154$$

Subtract 154 from both sides.

$$77+18x+x^{2}-154=0$$

Subtract 154 from 77 to get -77.

$$-77+18x+x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+18x-77=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 18 for b, and -77 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-77\right)}}{2}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-77\right)}}{2}$$

Multiply -4 times -77.

$$x=\frac{-18±\sqrt{324+308}}{2}$$

Add 324 to 308.

$$x=\frac{-18±\sqrt{632}}{2}$$

Take the square root of 632.

$$x=\frac{-18±2\sqrt{158}}{2}$$

Now solve the equation x=\frac{-18±2\sqrt{158}}{2} when ± is plus. Add -18 to 2\sqrt{158}.

$$x=\frac{2\sqrt{158}-18}{2}$$

Divide -18+2\sqrt{158} by 2.

$$x=\sqrt{158}-9$$

Now solve the equation x=\frac{-18±2\sqrt{158}}{2} when ± is minus. Subtract 2\sqrt{158} from -18.

$$x=\frac{-2\sqrt{158}-18}{2}$$

Divide -18-2\sqrt{158} by 2.

$$x=-\sqrt{158}-9$$

The equation is now solved.

$$x=\sqrt{158}-9 x=-\sqrt{158}-9$$

Steps for Completing the Square

Use the distributive property to multiply 11+x by 7+x and combine like terms.

$$154=77+18x+x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$77+18x+x^{2}=154$$

Subtract 77 from both sides.

$$18x+x^{2}=154-77$$

Subtract 77 from 154 to get 77.

$$18x+x^{2}=77$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+18x=77$$

Divide 18, the coefficient of the x term, by 2 to get 9. Then add the square of 9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+18x+9^{2}=77+9^{2}$$

Square 9.

$$x^{2}+18x+81=77+81$$

Add 77 to 81.

$$x^{2}+18x+81=158$$

Factor x^{2}+18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+9\right)^{2}=158$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{158}$$

Simplify.

$$x+9=\sqrt{158} x+9=-\sqrt{158}$$

Subtract 9 from both sides of the equation.

$$x=\sqrt{158}-9 x=-\sqrt{158}-9$$