Solve x^2-42x=759

$$x ^ { 2 } - 42 x = 759$$

Solve for x

$$x=20\sqrt{3}+21\approx 55.641016151$$

$$x=21-20\sqrt{3}\approx -13.641016151$$

Steps Using the Quadratic Formula

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.

$$x^{2}-42x=759$$

Subtract 759 from both sides of the equation.

$$x^{2}-42x-759=759-759$$

Subtracting 759 from itself leaves 0.

$$x^{2}-42x-759=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -42 for b, and -759 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-42\right)±\sqrt{\left(-42\right)^{2}-4\left(-759\right)}}{2}$$

Square -42.

$$x=\frac{-\left(-42\right)±\sqrt{1764-4\left(-759\right)}}{2}$$

Multiply -4 times -759.

$$x=\frac{-\left(-42\right)±\sqrt{1764+3036}}{2}$$

Add 1764 to 3036.

$$x=\frac{-\left(-42\right)±\sqrt{4800}}{2}$$

Take the square root of 4800.

$$x=\frac{-\left(-42\right)±40\sqrt{3}}{2}$$

The opposite of -42 is 42.

$$x=\frac{42±40\sqrt{3}}{2}$$

Now solve the equation x=\frac{42±40\sqrt{3}}{2} when ± is plus. Add 42 to 40\sqrt{3}.

$$x=\frac{40\sqrt{3}+42}{2}$$

Divide 42+40\sqrt{3} by 2.

$$x=20\sqrt{3}+21$$

Now solve the equation x=\frac{42±40\sqrt{3}}{2} when ± is minus. Subtract 40\sqrt{3} from 42.

$$x=\frac{42-40\sqrt{3}}{2}$$

Divide 42-40\sqrt{3} by 2.

$$x=21-20\sqrt{3}$$

The equation is now solved.

$$x=20\sqrt{3}+21 x=21-20\sqrt{3}$$

Steps for Completing the Square

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.

$$x^{2}-42x=759$$

Divide -42, the coefficient of the x term, by 2 to get -21. Then add the square of -21 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-42x+\left(-21\right)^{2}=759+\left(-21\right)^{2}$$

Square -21.

$$x^{2}-42x+441=759+441$$

Add 759 to 441.

$$x^{2}-42x+441=1200$$

Factor x^{2}-42x+441. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-21\right)^{2}=1200$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-21\right)^{2}}=\sqrt{1200}$$

Simplify.

$$x-21=20\sqrt{3} x-21=-20\sqrt{3}$$

Add 21 to both sides of the equation.

$$x=20\sqrt{3}+21 x=21-20\sqrt{3}$$