Solve x^2+2584=106x

$${ x }^{ 2 } +2584=106x$$

Solve for x

$$x=38$$

$$x=68$$

Steps Using the Quadratic Formula

Subtract 106x from both sides.

$$x^{2}+2584-106x=0$$

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.

$$x^{2}-106x+2584=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -106 for b, and 2584 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-106\right)±\sqrt{\left(-106\right)^{2}-4\times 2584}}{2}$$

Square -106.

$$x=\frac{-\left(-106\right)±\sqrt{11236-4\times 2584}}{2}$$

Multiply -4 times 2584.

$$x=\frac{-\left(-106\right)±\sqrt{11236-10336}}{2}$$

Add 11236 to -10336.

$$x=\frac{-\left(-106\right)±\sqrt{900}}{2}$$

Take the square root of 900.

$$x=\frac{-\left(-106\right)±30}{2}$$

The opposite of -106 is 106.

$$x=\frac{106±30}{2}$$

Now solve the equation x=\frac{106±30}{2} when ± is plus. Add 106 to 30.

$$x=\frac{136}{2}$$

Divide 136 by 2.

$$x=68$$

Now solve the equation x=\frac{106±30}{2} when ± is minus. Subtract 30 from 106.

$$x=\frac{76}{2}$$

Divide 76 by 2.

$$x=38$$

The equation is now solved.

$$x=68 x=38$$

Steps for Completing the Square

Subtract 106x from both sides.

$$x^{2}+2584-106x=0$$

Subtract 2584 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-106x=-2584$$

Divide -106, the coefficient of the x term, by 2 to get -53. Then add the square of -53 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-106x+\left(-53\right)^{2}=-2584+\left(-53\right)^{2}$$

Square -53.

$$x^{2}-106x+2809=-2584+2809$$

Add -2584 to 2809.

$$x^{2}-106x+2809=225$$

Factor x^{2}-106x+2809. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-53\right)^{2}=225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-53\right)^{2}}=\sqrt{225}$$

Simplify.

$$x-53=15 x-53=-15$$

Add 53 to both sides of the equation.

$$x=68 x=38$$