Solve 4078379=4040a-a^2

$$4078379 = 4040 a - a ^ { 2 }$$

Solve for a

$$a=\sqrt{2021}+2020\approx 2064.955533586$$

$$a=2020-\sqrt{2021}\approx 1975.044466414$$

Steps Using the Quadratic Formula

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

$$4040a-a^{2}=4078379$$

Subtract 4078379 from both sides.

$$4040a-a^{2}-4078379=0$$

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.

$$-a^{2}+4040a-4078379=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4040 for b, and -4078379 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-4040±\sqrt{4040^{2}-4\left(-1\right)\left(-4078379\right)}}{2\left(-1\right)}$$

Square 4040.

$$a=\frac{-4040±\sqrt{16321600-4\left(-1\right)\left(-4078379\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-4040±\sqrt{16321600+4\left(-4078379\right)}}{2\left(-1\right)}$$

Multiply 4 times -4078379.

$$a=\frac{-4040±\sqrt{16321600-16313516}}{2\left(-1\right)}$$

Add 16321600 to -16313516.

$$a=\frac{-4040±\sqrt{8084}}{2\left(-1\right)}$$

Take the square root of 8084.

$$a=\frac{-4040±2\sqrt{2021}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{-4040±2\sqrt{2021}}{-2}$$

Now solve the equation a=\frac{-4040±2\sqrt{2021}}{-2} when ± is plus. Add -4040 to 2\sqrt{2021}.

$$a=\frac{2\sqrt{2021}-4040}{-2}$$

Divide -4040+2\sqrt{2021} by -2.

$$a=2020-\sqrt{2021}$$

Now solve the equation a=\frac{-4040±2\sqrt{2021}}{-2} when ± is minus. Subtract 2\sqrt{2021} from -4040.

$$a=\frac{-2\sqrt{2021}-4040}{-2}$$

Divide -4040-2\sqrt{2021} by -2.

$$a=\sqrt{2021}+2020$$

The equation is now solved.

$$a=2020-\sqrt{2021} a=\sqrt{2021}+2020$$

Steps for Completing the Square

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

$$4040a-a^{2}=4078379$$

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.

$$-a^{2}+4040a=4078379$$

Divide both sides by -1.

$$\frac{-a^{2}+4040a}{-1}=\frac{4078379}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$a^{2}+\frac{4040}{-1}a=\frac{4078379}{-1}$$

Divide 4040 by -1.

$$a^{2}-4040a=\frac{4078379}{-1}$$

Divide 4078379 by -1.

$$a^{2}-4040a=-4078379$$

Divide -4040, the coefficient of the x term, by 2 to get -2020. Then add the square of -2020 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-4040a+\left(-2020\right)^{2}=-4078379+\left(-2020\right)^{2}$$

Square -2020.

$$a^{2}-4040a+4080400=-4078379+4080400$$

Add -4078379 to 4080400.

$$a^{2}-4040a+4080400=2021$$

Factor a^{2}-4040a+4080400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-2020\right)^{2}=2021$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-2020\right)^{2}}=\sqrt{2021}$$

Simplify.

$$a-2020=\sqrt{2021} a-2020=-\sqrt{2021}$$

Add 2020 to both sides of the equation.

$$a=\sqrt{2021}+2020 a=2020-\sqrt{2021}$$