Solve 2(a+3)+a(a-6)+35=a(a-5)+a(a+7)=

$$2 ( a + 3 ) + a ( a - 6 ) + 35 = a ( a - 5 ) + a ( a + 7 ) =$$

Solve for a

$$a=5\sqrt{2}-3\approx 4.071067812$$

$$a=-5\sqrt{2}-3\approx -10.071067812$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2 by a+3.

$$2a+6+a\left(a-6\right)+35=a\left(a-5\right)+a\left(a+7\right)$$

Use the distributive property to multiply a by a-6.

$$2a+6+a^{2}-6a+35=a\left(a-5\right)+a\left(a+7\right)$$

Combine 2a and -6a to get -4a.

$$-4a+6+a^{2}+35=a\left(a-5\right)+a\left(a+7\right)$$

Add 6 and 35 to get 41.

$$-4a+41+a^{2}=a\left(a-5\right)+a\left(a+7\right)$$

Use the distributive property to multiply a by a-5.

$$-4a+41+a^{2}=a^{2}-5a+a\left(a+7\right)$$

Use the distributive property to multiply a by a+7.

$$-4a+41+a^{2}=a^{2}-5a+a^{2}+7a$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$-4a+41+a^{2}=2a^{2}-5a+7a$$

Combine -5a and 7a to get 2a.

$$-4a+41+a^{2}=2a^{2}+2a$$

Subtract 2a^{2} from both sides.

$$-4a+41+a^{2}-2a^{2}=2a$$

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

$$-4a+41-a^{2}=2a$$

Subtract 2a from both sides.

$$-4a+41-a^{2}-2a=0$$

Combine -4a and -2a to get -6a.

$$-6a+41-a^{2}=0$$

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.

$$-a^{2}-6a+41=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -6 for b, and 41 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\times 41}}{2\left(-1\right)}$$

Square -6.

$$a=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\times 41}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-\left(-6\right)±\sqrt{36+4\times 41}}{2\left(-1\right)}$$

Multiply 4 times 41.

$$a=\frac{-\left(-6\right)±\sqrt{36+164}}{2\left(-1\right)}$$

Add 36 to 164.

$$a=\frac{-\left(-6\right)±\sqrt{200}}{2\left(-1\right)}$$

Take the square root of 200.

$$a=\frac{-\left(-6\right)±10\sqrt{2}}{2\left(-1\right)}$$

The opposite of -6 is 6.

$$a=\frac{6±10\sqrt{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{6±10\sqrt{2}}{-2}$$

Now solve the equation a=\frac{6±10\sqrt{2}}{-2} when ± is plus. Add 6 to 10\sqrt{2}.

$$a=\frac{10\sqrt{2}+6}{-2}$$

Divide 6+10\sqrt{2} by -2.

$$a=-5\sqrt{2}-3$$

Now solve the equation a=\frac{6±10\sqrt{2}}{-2} when ± is minus. Subtract 10\sqrt{2} from 6.

$$a=\frac{6-10\sqrt{2}}{-2}$$

Divide 6-10\sqrt{2} by -2.

$$a=5\sqrt{2}-3$$

The equation is now solved.

$$a=-5\sqrt{2}-3 a=5\sqrt{2}-3$$

Steps for Completing the Square

Use the distributive property to multiply 2 by a+3.

$$2a+6+a\left(a-6\right)+35=a\left(a-5\right)+a\left(a+7\right)$$

Use the distributive property to multiply a by a-6.

$$2a+6+a^{2}-6a+35=a\left(a-5\right)+a\left(a+7\right)$$

Combine 2a and -6a to get -4a.

$$-4a+6+a^{2}+35=a\left(a-5\right)+a\left(a+7\right)$$

Add 6 and 35 to get 41.

$$-4a+41+a^{2}=a\left(a-5\right)+a\left(a+7\right)$$

Use the distributive property to multiply a by a-5.

$$-4a+41+a^{2}=a^{2}-5a+a\left(a+7\right)$$

Use the distributive property to multiply a by a+7.

$$-4a+41+a^{2}=a^{2}-5a+a^{2}+7a$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$-4a+41+a^{2}=2a^{2}-5a+7a$$

Combine -5a and 7a to get 2a.

$$-4a+41+a^{2}=2a^{2}+2a$$

Subtract 2a^{2} from both sides.

$$-4a+41+a^{2}-2a^{2}=2a$$

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

$$-4a+41-a^{2}=2a$$

Subtract 2a from both sides.

$$-4a+41-a^{2}-2a=0$$

Combine -4a and -2a to get -6a.

$$-6a+41-a^{2}=0$$

Subtract 41 from both sides. Anything subtracted from zero gives its negation.

$$-6a-a^{2}=-41$$

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.

$$-a^{2}-6a=-41$$

Divide both sides by -1.

$$\frac{-a^{2}-6a}{-1}=-\frac{41}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$a^{2}+\left(-\frac{6}{-1}\right)a=-\frac{41}{-1}$$

Divide -6 by -1.

$$a^{2}+6a=-\frac{41}{-1}$$

Divide -41 by -1.

$$a^{2}+6a=41$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+6a+3^{2}=41+3^{2}$$

Square 3.

$$a^{2}+6a+9=41+9$$

Add 41 to 9.

$$a^{2}+6a+9=50$$

Factor a^{2}+6a+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+3\right)^{2}=50$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+3\right)^{2}}=\sqrt{50}$$

Simplify.

$$a+3=5\sqrt{2} a+3=-5\sqrt{2}$$

Subtract 3 from both sides of the equation.

$$a=5\sqrt{2}-3 a=-5\sqrt{2}-3$$