Solve left(7+xright)left(3-xright)=5

$$\left( 7+x \right) \left( 3-x \right) =5$$

Solve for x

$$x=2\sqrt{5}-2\approx 2.472135955$$

$$x=-2\sqrt{5}-2\approx -6.472135955$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 7+x by 3-x and combine like terms.

$$21-4x-x^{2}=5$$

Subtract 5 from both sides.

$$21-4x-x^{2}-5=0$$

Subtract 5 from 21 to get 16.

$$16-4x-x^{2}=0$$

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.

$$-x^{2}-4x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -4 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 16}}{2\left(-1\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 16}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\times 16}}{2\left(-1\right)}$$

Multiply 4 times 16.

$$x=\frac{-\left(-4\right)±\sqrt{16+64}}{2\left(-1\right)}$$

Add 16 to 64.

$$x=\frac{-\left(-4\right)±\sqrt{80}}{2\left(-1\right)}$$

Take the square root of 80.

$$x=\frac{-\left(-4\right)±4\sqrt{5}}{2\left(-1\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{4±4\sqrt{5}}{-2}$$

Now solve the equation x=\frac{4±4\sqrt{5}}{-2} when ± is plus. Add 4 to 4\sqrt{5}.

$$x=\frac{4\sqrt{5}+4}{-2}$$

Divide 4+4\sqrt{5} by -2.

$$x=-2\sqrt{5}-2$$

Now solve the equation x=\frac{4±4\sqrt{5}}{-2} when ± is minus. Subtract 4\sqrt{5} from 4.

$$x=\frac{4-4\sqrt{5}}{-2}$$

Divide 4-4\sqrt{5} by -2.

$$x=2\sqrt{5}-2$$

The equation is now solved.

$$x=-2\sqrt{5}-2 x=2\sqrt{5}-2$$

Steps for Completing the Square

Use the distributive property to multiply 7+x by 3-x and combine like terms.

$$21-4x-x^{2}=5$$

Subtract 21 from both sides.

$$-4x-x^{2}=5-21$$

Subtract 21 from 5 to get -16.

$$-4x-x^{2}=-16$$

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.

$$-x^{2}-4x=-16$$

Divide both sides by -1.

$$\frac{-x^{2}-4x}{-1}=-\frac{16}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{4}{-1}\right)x=-\frac{16}{-1}$$

Divide -4 by -1.

$$x^{2}+4x=-\frac{16}{-1}$$

Divide -16 by -1.

$$x^{2}+4x=16$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4x+2^{2}=16+2^{2}$$

Square 2.

$$x^{2}+4x+4=16+4$$

Add 16 to 4.

$$x^{2}+4x+4=20$$

Factor x^{2}+4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+2\right)^{2}=20$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{20}$$

Simplify.

$$x+2=2\sqrt{5} x+2=-2\sqrt{5}$$

Subtract 2 from both sides of the equation.

$$x=2\sqrt{5}-2 x=-2\sqrt{5}-2$$