Solve 2943=a^2+1/2a*41*4

$$2943 = a ^ { 2 } + \frac { 1 } { 2 } a \cdot 41 \cdot 4$$

Solve for a

$$a=-109$$

$$a=27$$

Steps Using Factoring

Multiply \frac{1}{2} and 41 to get \frac{41}{2}.

$$2943=a^{2}+\frac{41}{2}a\times 4$$

Multiply \frac{41}{2} and 4 to get 82.

$$2943=a^{2}+82a$$

Swap sides so that all variable terms are on the left hand side.

$$a^{2}+82a=2943$$

Subtract 2943 from both sides.

$$a^{2}+82a-2943=0$$

To solve the equation, factor a^{2}+82a-2943 using formula a^{2}+\left(a+b\right)a+ab=\left(a+a\right)\left(a+b\right). To find a and b, set up a system to be solved.

$$a+b=82 ab=-2943$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -2943.

$$-1,2943 -3,981 -9,327 -27,109$$

Calculate the sum for each pair.

$$-1+2943=2942 -3+981=978 -9+327=318 -27+109=82$$

The solution is the pair that gives sum 82.

$$a=-27 b=109$$

Rewrite factored expression \left(a+a\right)\left(a+b\right) using the obtained values.

$$\left(a-27\right)\left(a+109\right)$$

To find equation solutions, solve a-27=0 and a+109=0.

$$a=27 a=-109$$

Steps Using Factoring By Grouping

Multiply \frac{1}{2} and 41 to get \frac{41}{2}.

$$2943=a^{2}+\frac{41}{2}a\times 4$$

Multiply \frac{41}{2} and 4 to get 82.

$$2943=a^{2}+82a$$

Swap sides so that all variable terms are on the left hand side.

$$a^{2}+82a=2943$$

Subtract 2943 from both sides.

$$a^{2}+82a-2943=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as a^{2}+aa+ba-2943. To find a and b, set up a system to be solved.

$$a+b=82 ab=1\left(-2943\right)=-2943$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -2943.

$$-1,2943 -3,981 -9,327 -27,109$$

Calculate the sum for each pair.

$$-1+2943=2942 -3+981=978 -9+327=318 -27+109=82$$

The solution is the pair that gives sum 82.

$$a=-27 b=109$$

Rewrite a^{2}+82a-2943 as \left(a^{2}-27a\right)+\left(109a-2943\right).

$$\left(a^{2}-27a\right)+\left(109a-2943\right)$$

Factor out a in the first and 109 in the second group.

$$a\left(a-27\right)+109\left(a-27\right)$$

Factor out common term a-27 by using distributive property.

$$\left(a-27\right)\left(a+109\right)$$

To find equation solutions, solve a-27=0 and a+109=0.

$$a=27 a=-109$$

Steps Using the Quadratic Formula

Multiply \frac{1}{2} and 41 to get \frac{41}{2}.

$$2943=a^{2}+\frac{41}{2}a\times 4$$

Multiply \frac{41}{2} and 4 to get 82.

$$2943=a^{2}+82a$$

Swap sides so that all variable terms are on the left hand side.

$$a^{2}+82a=2943$$

Subtract 2943 from both sides.

$$a^{2}+82a-2943=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 82 for b, and -2943 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-82±\sqrt{82^{2}-4\left(-2943\right)}}{2}$$

Square 82.

$$a=\frac{-82±\sqrt{6724-4\left(-2943\right)}}{2}$$

Multiply -4 times -2943.

$$a=\frac{-82±\sqrt{6724+11772}}{2}$$

Add 6724 to 11772.

$$a=\frac{-82±\sqrt{18496}}{2}$$

Take the square root of 18496.

$$a=\frac{-82±136}{2}$$

Now solve the equation a=\frac{-82±136}{2} when ± is plus. Add -82 to 136.

$$a=\frac{54}{2}$$

Divide 54 by 2.

$$a=27$$

Now solve the equation a=\frac{-82±136}{2} when ± is minus. Subtract 136 from -82.

$$a=-\frac{218}{2}$$

Divide -218 by 2.

$$a=-109$$

The equation is now solved.

$$a=27 a=-109$$

Steps for Completing the Square

Multiply \frac{1}{2} and 41 to get \frac{41}{2}.

$$2943=a^{2}+\frac{41}{2}a\times 4$$

Multiply \frac{41}{2} and 4 to get 82.

$$2943=a^{2}+82a$$

Swap sides so that all variable terms are on the left hand side.

$$a^{2}+82a=2943$$

Divide 82, the coefficient of the x term, by 2 to get 41. Then add the square of 41 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+82a+41^{2}=2943+41^{2}$$

Square 41.

$$a^{2}+82a+1681=2943+1681$$

Add 2943 to 1681.

$$a^{2}+82a+1681=4624$$

Factor a^{2}+82a+1681. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+41\right)^{2}=4624$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+41\right)^{2}}=\sqrt{4624}$$

Simplify.

$$a+41=68 a+41=-68$$

Subtract 41 from both sides of the equation.

$$a=27 a=-109$$