Solve (16+x)(14+x)=299

$$( 16 + x ) ( 14 + x ) = 299$$

Solve for x

$$x=10\sqrt{3}-15\approx 2.320508076$$

$$x=-10\sqrt{3}-15\approx -32.320508076$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 16+x by 14+x and combine like terms.

$$224+30x+x^{2}=299$$

Subtract 299 from both sides.

$$224+30x+x^{2}-299=0$$

Subtract 299 from 224 to get -75.

$$-75+30x+x^{2}=0$$

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.

$$x^{2}+30x-75=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 30 for b, and -75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-75\right)}}{2}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\left(-75\right)}}{2}$$

Multiply -4 times -75.

$$x=\frac{-30±\sqrt{900+300}}{2}$$

Add 900 to 300.

$$x=\frac{-30±\sqrt{1200}}{2}$$

Take the square root of 1200.

$$x=\frac{-30±20\sqrt{3}}{2}$$

Now solve the equation x=\frac{-30±20\sqrt{3}}{2} when ± is plus. Add -30 to 20\sqrt{3}.

$$x=\frac{20\sqrt{3}-30}{2}$$

Divide -30+20\sqrt{3} by 2.

$$x=10\sqrt{3}-15$$

Now solve the equation x=\frac{-30±20\sqrt{3}}{2} when ± is minus. Subtract 20\sqrt{3} from -30.

$$x=\frac{-20\sqrt{3}-30}{2}$$

Divide -30-20\sqrt{3} by 2.

$$x=-10\sqrt{3}-15$$

The equation is now solved.

$$x=10\sqrt{3}-15 x=-10\sqrt{3}-15$$

Steps for Completing the Square

Use the distributive property to multiply 16+x by 14+x and combine like terms.

$$224+30x+x^{2}=299$$

Subtract 224 from both sides.

$$30x+x^{2}=299-224$$

Subtract 224 from 299 to get 75.

$$30x+x^{2}=75$$

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.

$$x^{2}+30x=75$$

Divide 30, the coefficient of the x term, by 2 to get 15. Then add the square of 15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+30x+15^{2}=75+15^{2}$$

Square 15.

$$x^{2}+30x+225=75+225$$

Add 75 to 225.

$$x^{2}+30x+225=300$$

Factor x^{2}+30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+15\right)^{2}=300$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+15\right)^{2}}=\sqrt{300}$$

Simplify.

$$x+15=10\sqrt{3} x+15=-10\sqrt{3}$$

Subtract 15 from both sides of the equation.

$$x=10\sqrt{3}-15 x=-10\sqrt{3}-15$$