Solve 4x+5+x^2=2x+6-104x^2

$$4 x + 5 + x ^ { 2 } = 2 x + 6 - 104 x ^ { 2 }$$

Solve for x

$$x=\frac{\sqrt{106}-1}{105}\approx 0.088529811$$

$$x=\frac{-\sqrt{106}-1}{105}\approx -0.10757743$$

Steps Using the Quadratic Formula

Subtract 2x from both sides.

$$4x+5+x^{2}-2x=6-104x^{2}$$

Combine 4x and -2x to get 2x.

$$2x+5+x^{2}=6-104x^{2}$$

Subtract 6 from both sides.

$$2x+5+x^{2}-6=-104x^{2}$$

Subtract 6 from 5 to get -1.

$$2x-1+x^{2}=-104x^{2}$$

Add 104x^{2} to both sides.

$$2x-1+x^{2}+104x^{2}=0$$

Combine x^{2} and 104x^{2} to get 105x^{2}.

$$2x-1+105x^{2}=0$$

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.

$$105x^{2}+2x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 105 for a, 2 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 105\left(-1\right)}}{2\times 105}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 105\left(-1\right)}}{2\times 105}$$

Multiply -4 times 105.

$$x=\frac{-2±\sqrt{4-420\left(-1\right)}}{2\times 105}$$

Multiply -420 times -1.

$$x=\frac{-2±\sqrt{4+420}}{2\times 105}$$

Add 4 to 420.

$$x=\frac{-2±\sqrt{424}}{2\times 105}$$

Take the square root of 424.

$$x=\frac{-2±2\sqrt{106}}{2\times 105}$$

Multiply 2 times 105.

$$x=\frac{-2±2\sqrt{106}}{210}$$

Now solve the equation x=\frac{-2±2\sqrt{106}}{210} when ± is plus. Add -2 to 2\sqrt{106}.

$$x=\frac{2\sqrt{106}-2}{210}$$

Divide -2+2\sqrt{106} by 210.

$$x=\frac{\sqrt{106}-1}{105}$$

Now solve the equation x=\frac{-2±2\sqrt{106}}{210} when ± is minus. Subtract 2\sqrt{106} from -2.

$$x=\frac{-2\sqrt{106}-2}{210}$$

Divide -2-2\sqrt{106} by 210.

$$x=\frac{-\sqrt{106}-1}{105}$$

The equation is now solved.

$$x=\frac{\sqrt{106}-1}{105} x=\frac{-\sqrt{106}-1}{105}$$

Steps for Completing the Square

Subtract 2x from both sides.

$$4x+5+x^{2}-2x=6-104x^{2}$$

Combine 4x and -2x to get 2x.

$$2x+5+x^{2}=6-104x^{2}$$

Add 104x^{2} to both sides.

$$2x+5+x^{2}+104x^{2}=6$$

Combine x^{2} and 104x^{2} to get 105x^{2}.

$$2x+5+105x^{2}=6$$

Subtract 5 from both sides.

$$2x+105x^{2}=6-5$$

Subtract 5 from 6 to get 1.

$$2x+105x^{2}=1$$

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.

$$105x^{2}+2x=1$$

Divide both sides by 105.

$$\frac{105x^{2}+2x}{105}=\frac{1}{105}$$

Dividing by 105 undoes the multiplication by 105.

$$x^{2}+\frac{2}{105}x=\frac{1}{105}$$

Divide \frac{2}{105}, the coefficient of the x term, by 2 to get \frac{1}{105}. Then add the square of \frac{1}{105} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{2}{105}x+\left(\frac{1}{105}\right)^{2}=\frac{1}{105}+\left(\frac{1}{105}\right)^{2}$$

Square \frac{1}{105} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{2}{105}x+\frac{1}{11025}=\frac{1}{105}+\frac{1}{11025}$$

Add \frac{1}{105} to \frac{1}{11025} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{2}{105}x+\frac{1}{11025}=\frac{106}{11025}$$

Factor x^{2}+\frac{2}{105}x+\frac{1}{11025}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{105}\right)^{2}=\frac{106}{11025}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{105}\right)^{2}}=\sqrt{\frac{106}{11025}}$$

Simplify.

$$x+\frac{1}{105}=\frac{\sqrt{106}}{105} x+\frac{1}{105}=-\frac{\sqrt{106}}{105}$$

Subtract \frac{1}{105} from both sides of the equation.

$$x=\frac{\sqrt{106}-1}{105} x=\frac{-\sqrt{106}-1}{105}$$