Solve (a-7)(a-5)=3

$$( a - 7 ) ( a - 5 ) = 3$$

Solve for a

$$a=4$$

$$a=8$$

Steps Using the Quadratic Formula

Use the distributive property to multiply a-7 by a-5 and combine like terms.

$$a^{2}-12a+35=3$$

Subtract 3 from both sides.

$$a^{2}-12a+35-3=0$$

Subtract 3 from 35 to get 32.

$$a^{2}-12a+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 32}}{2}$$

Square -12.

$$a=\frac{-\left(-12\right)±\sqrt{144-4\times 32}}{2}$$

Multiply -4 times 32.

$$a=\frac{-\left(-12\right)±\sqrt{144-128}}{2}$$

Add 144 to -128.

$$a=\frac{-\left(-12\right)±\sqrt{16}}{2}$$

Take the square root of 16.

$$a=\frac{-\left(-12\right)±4}{2}$$

The opposite of -12 is 12.

$$a=\frac{12±4}{2}$$

Now solve the equation a=\frac{12±4}{2} when ± is plus. Add 12 to 4.

$$a=\frac{16}{2}$$

Divide 16 by 2.

$$a=8$$

Now solve the equation a=\frac{12±4}{2} when ± is minus. Subtract 4 from 12.

$$a=\frac{8}{2}$$

Divide 8 by 2.

$$a=4$$

The equation is now solved.

$$a=8 a=4$$

Steps for Completing the Square

Use the distributive property to multiply a-7 by a-5 and combine like terms.

$$a^{2}-12a+35=3$$

Subtract 35 from both sides.

$$a^{2}-12a=3-35$$

Subtract 35 from 3 to get -32.

$$a^{2}-12a=-32$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-12a+\left(-6\right)^{2}=-32+\left(-6\right)^{2}$$

Square -6.

$$a^{2}-12a+36=-32+36$$

Add -32 to 36.

$$a^{2}-12a+36=4$$

Factor a^{2}-12a+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-6\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-6\right)^{2}}=\sqrt{4}$$

Simplify.

$$a-6=2 a-6=-2$$

Add 6 to both sides of the equation.

$$a=8 a=4$$