Solve x^2+x+1=25x+2

$$x ^ { 2 } + x + 1 = 25 x + 2$$

Solve for x

$$x=\sqrt{145}+12\approx 24.041594579$$

$$x=12-\sqrt{145}\approx -0.041594579$$

Steps Using the Quadratic Formula

Subtract 25x from both sides.

$$x^{2}+x+1-25x=2$$

Combine x and -25x to get -24x.

$$x^{2}-24x+1=2$$

Subtract 2 from both sides.

$$x^{2}-24x+1-2=0$$

Subtract 2 from 1 to get -1.

$$x^{2}-24x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -24 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\left(-1\right)}}{2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$x=\frac{-\left(-24\right)±\sqrt{576+4}}{2}$$

Add 576 to 4.

$$x=\frac{-\left(-24\right)±\sqrt{580}}{2}$$

Take the square root of 580.

$$x=\frac{-\left(-24\right)±2\sqrt{145}}{2}$$

The opposite of -24 is 24.

$$x=\frac{24±2\sqrt{145}}{2}$$

Now solve the equation x=\frac{24±2\sqrt{145}}{2} when ± is plus. Add 24 to 2\sqrt{145}.

$$x=\frac{2\sqrt{145}+24}{2}$$

Divide 24+2\sqrt{145} by 2.

$$x=\sqrt{145}+12$$

Now solve the equation x=\frac{24±2\sqrt{145}}{2} when ± is minus. Subtract 2\sqrt{145} from 24.

$$x=\frac{24-2\sqrt{145}}{2}$$

Divide 24-2\sqrt{145} by 2.

$$x=12-\sqrt{145}$$

The equation is now solved.

$$x=\sqrt{145}+12 x=12-\sqrt{145}$$

Steps for Completing the Square

Subtract 25x from both sides.

$$x^{2}+x+1-25x=2$$

Combine x and -25x to get -24x.

$$x^{2}-24x+1=2$$

Subtract 1 from both sides.

$$x^{2}-24x=2-1$$

Subtract 1 from 2 to get 1.

$$x^{2}-24x=1$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-24x+\left(-12\right)^{2}=1+\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=1+144$$

Add 1 to 144.

$$x^{2}-24x+144=145$$

Factor x^{2}-24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-12\right)^{2}=145$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{145}$$

Simplify.

$$x-12=\sqrt{145} x-12=-\sqrt{145}$$

Add 12 to both sides of the equation.

$$x=\sqrt{145}+12 x=12-\sqrt{145}$$