Solve 4x+4,8+2,4x=x^2+2x

$$4 x + 4,8 + 2,4 x = x ^ { 2 } + 2 x$$

Solve for x

$$x = \frac{\sqrt{241} + 11}{5} \approx 5.304834939$$

$$x=\frac{11-\sqrt{241}}{5}\approx -0.904834939$$

Steps Using the Quadratic Formula

Combine 4x and 2,4x to get 6,4x.

$$6,4x+4,8=x^{2}+2x$$

Subtract x^{2} from both sides.

$$6,4x+4,8-x^{2}=2x$$

Subtract 2x from both sides.

$$6,4x+4,8-x^{2}-2x=0$$

Combine 6,4x and -2x to get 4,4x.

$$4,4x+4,8-x^{2}=0$$

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.

$$-x^{2}+4,4x+4,8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4,4 for b, and 4,8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4,4±\sqrt{4,4^{2}-4\left(-1\right)\times 4,8}}{2\left(-1\right)}$$

Square 4,4 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-4,4±\sqrt{19,36-4\left(-1\right)\times 4,8}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-4,4±\sqrt{19,36+4\times 4,8}}{2\left(-1\right)}$$

Multiply 4 times 4,8.

$$x=\frac{-4,4±\sqrt{19,36+19,2}}{2\left(-1\right)}$$

Add 19,36 to 19,2 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-4,4±\sqrt{38,56}}{2\left(-1\right)}$$

Take the square root of 38,56.

$$x=\frac{-4,4±\frac{2\sqrt{241}}{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-4,4±\frac{2\sqrt{241}}{5}}{-2}$$

Now solve the equation x=\frac{-4,4±\frac{2\sqrt{241}}{5}}{-2} when ± is plus. Add -4,4 to \frac{2\sqrt{241}}{5}.

$$x=\frac{2\sqrt{241}-22}{-2\times 5}$$

Divide \frac{-22+2\sqrt{241}}{5} by -2.

$$x=\frac{11-\sqrt{241}}{5}$$

Now solve the equation x=\frac{-4,4±\frac{2\sqrt{241}}{5}}{-2} when ± is minus. Subtract \frac{2\sqrt{241}}{5} from -4,4.

$$x=\frac{-2\sqrt{241}-22}{-2\times 5}$$

Divide \frac{-22-2\sqrt{241}}{5} by -2.

$$x=\frac{\sqrt{241}+11}{5}$$

The equation is now solved.

$$x=\frac{11-\sqrt{241}}{5} x=\frac{\sqrt{241}+11}{5}$$

Steps for Completing the Square

Combine 4x and 2,4x to get 6,4x.

$$6,4x+4,8=x^{2}+2x$$

Subtract x^{2} from both sides.

$$6,4x+4,8-x^{2}=2x$$

Subtract 2x from both sides.

$$6,4x+4,8-x^{2}-2x=0$$

Combine 6,4x and -2x to get 4,4x.

$$4,4x+4,8-x^{2}=0$$

Subtract 4,8 from both sides. Anything subtracted from zero gives its negation.

$$4,4x-x^{2}=-4,8$$

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.

$$-x^{2}+4,4x=-4,8$$

Divide both sides by -1.

$$\frac{-x^{2}+4,4x}{-1}=-\frac{4,8}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{4,4}{-1}x=-\frac{4,8}{-1}$$

Divide 4,4 by -1.

$$x^{2}-4,4x=-\frac{4,8}{-1}$$

Divide -4,8 by -1.

$$x^{2}-4,4x=4,8$$

Divide -4,4, the coefficient of the x term, by 2 to get -2,2. Then add the square of -2,2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4,4x+\left(-2,2\right)^{2}=4,8+\left(-2,2\right)^{2}$$

Square -2,2 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-4,4x+4,84=4,8+4,84$$

Add 4,8 to 4,84 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-4,4x+4,84=9,64$$

Factor x^{2}-4,4x+4,84. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2,2\right)^{2}=9,64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2,2\right)^{2}}=\sqrt{9,64}$$

Simplify.

$$x-2,2=\frac{\sqrt{241}}{5} x-2,2=-\frac{\sqrt{241}}{5}$$

Add 2,2 to both sides of the equation.

$$x=\frac{\sqrt{241}+11}{5} x=\frac{11-\sqrt{241}}{5}$$