Solve (7+z)(9-z)+(7-z)(9+z)=76

$$( 7 + z ) ( 9 - z ) + ( 7 - z ) ( 9 + z ) = 76$$

Solve for z

$$z=5$$

$$z=-5$$

Steps by Finding Square Root

Use the distributive property to multiply 7+z by 9-z and combine like terms.

$$63+2z-z^{2}+\left(7-z\right)\left(9+z\right)=76$$

Use the distributive property to multiply 7-z by 9+z and combine like terms.

$$63+2z-z^{2}+63-2z-z^{2}=76$$

Add 63 and 63 to get 126.

$$126+2z-z^{2}-2z-z^{2}=76$$

Combine 2z and -2z to get 0.

$$126-z^{2}-z^{2}=76$$

Combine -z^{2} and -z^{2} to get -2z^{2}.

$$126-2z^{2}=76$$

Subtract 126 from both sides.

$$-2z^{2}=76-126$$

Subtract 126 from 76 to get -50.

$$-2z^{2}=-50$$

Divide both sides by -2.

$$z^{2}=\frac{-50}{-2}$$

Divide -50 by -2 to get 25.

$$z^{2}=25$$

Take the square root of both sides of the equation.

$$z=5 z=-5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 7+z by 9-z and combine like terms.

$$63+2z-z^{2}+\left(7-z\right)\left(9+z\right)=76$$

Use the distributive property to multiply 7-z by 9+z and combine like terms.

$$63+2z-z^{2}+63-2z-z^{2}=76$$

Add 63 and 63 to get 126.

$$126+2z-z^{2}-2z-z^{2}=76$$

Combine 2z and -2z to get 0.

$$126-z^{2}-z^{2}=76$$

Combine -z^{2} and -z^{2} to get -2z^{2}.

$$126-2z^{2}=76$$

Subtract 76 from both sides.

$$126-2z^{2}-76=0$$

Subtract 76 from 126 to get 50.

$$50-2z^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$-2z^{2}+50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 0 for b, and 50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{0±\sqrt{0^{2}-4\left(-2\right)\times 50}}{2\left(-2\right)}$$

Square 0.

$$z=\frac{0±\sqrt{-4\left(-2\right)\times 50}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$z=\frac{0±\sqrt{8\times 50}}{2\left(-2\right)}$$

Multiply 8 times 50.

$$z=\frac{0±\sqrt{400}}{2\left(-2\right)}$$

Take the square root of 400.

$$z=\frac{0±20}{2\left(-2\right)}$$

Multiply 2 times -2.

$$z=\frac{0±20}{-4}$$

Now solve the equation z=\frac{0±20}{-4} when ± is plus. Divide 20 by -4.

$$z=-5$$

Now solve the equation z=\frac{0±20}{-4} when ± is minus. Divide -20 by -4.

$$z=5$$

The equation is now solved.

$$z=-5 z=5$$