Solve 14x+4,8+2,4x=x^2+2x

$$14 x + 4,8 + 2,4 x = x ^ { 2 } + 2 x$$

Solve for x

$$x = \frac{2 \sqrt{354} + 36}{5} \approx 14.725955089$$

$$x=\frac{36-2\sqrt{354}}{5}\approx -0.325955089$$

Steps Using the Quadratic Formula

Combine 14x and 2,4x to get 16,4x.

$$16,4x+4,8=x^{2}+2x$$

Subtract x^{2} from both sides.

$$16,4x+4,8-x^{2}=2x$$

Subtract 2x from both sides.

$$16,4x+4,8-x^{2}-2x=0$$

Combine 16,4x and -2x to get 14,4x.

$$14,4x+4,8-x^{2}=0$$

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.

$$-x^{2}+14,4x+4,8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 14,4 for b, and 4,8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14,4±\sqrt{14,4^{2}-4\left(-1\right)\times 4,8}}{2\left(-1\right)}$$

Square 14,4 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-14,4±\sqrt{207,36-4\left(-1\right)\times 4,8}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-14,4±\sqrt{207,36+4\times 4,8}}{2\left(-1\right)}$$

Multiply 4 times 4,8.

$$x=\frac{-14,4±\sqrt{207,36+19,2}}{2\left(-1\right)}$$

Add 207,36 to 19,2 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-14,4±\sqrt{226,56}}{2\left(-1\right)}$$

Take the square root of 226,56.

$$x=\frac{-14,4±\frac{4\sqrt{354}}{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-14,4±\frac{4\sqrt{354}}{5}}{-2}$$

Now solve the equation x=\frac{-14,4±\frac{4\sqrt{354}}{5}}{-2} when ± is plus. Add -14,4 to \frac{4\sqrt{354}}{5}.

$$x=\frac{4\sqrt{354}-72}{-2\times 5}$$

Divide \frac{-72+4\sqrt{354}}{5} by -2.

$$x=\frac{36-2\sqrt{354}}{5}$$

Now solve the equation x=\frac{-14,4±\frac{4\sqrt{354}}{5}}{-2} when ± is minus. Subtract \frac{4\sqrt{354}}{5} from -14,4.

$$x=\frac{-4\sqrt{354}-72}{-2\times 5}$$

Divide \frac{-72-4\sqrt{354}}{5} by -2.

$$x=\frac{2\sqrt{354}+36}{5}$$

The equation is now solved.

$$x=\frac{36-2\sqrt{354}}{5} x=\frac{2\sqrt{354}+36}{5}$$

Steps for Completing the Square

Combine 14x and 2,4x to get 16,4x.

$$16,4x+4,8=x^{2}+2x$$

Subtract x^{2} from both sides.

$$16,4x+4,8-x^{2}=2x$$

Subtract 2x from both sides.

$$16,4x+4,8-x^{2}-2x=0$$

Combine 16,4x and -2x to get 14,4x.

$$14,4x+4,8-x^{2}=0$$

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

$$14,4x-x^{2}=-4,8$$

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.

$$-x^{2}+14,4x=-4,8$$

Divide both sides by -1.

$$\frac{-x^{2}+14,4x}{-1}=-\frac{4,8}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{14,4}{-1}x=-\frac{4,8}{-1}$$

Divide 14,4 by -1.

$$x^{2}-14,4x=-\frac{4,8}{-1}$$

Divide -4,8 by -1.

$$x^{2}-14,4x=4,8$$

Divide -14,4, the coefficient of the x term, by 2 to get -7,2. Then add the square of -7,2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14,4x+\left(-7,2\right)^{2}=4,8+\left(-7,2\right)^{2}$$

Square -7,2 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-14,4x+51,84=4,8+51,84$$

Add 4,8 to 51,84 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-14,4x+51,84=56,64$$

Factor x^{2}-14,4x+51,84. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7,2\right)^{2}=56,64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7,2\right)^{2}}=\sqrt{56,64}$$

Simplify.

$$x-7,2=\frac{2\sqrt{354}}{5} x-7,2=-\frac{2\sqrt{354}}{5}$$

Add 7,2 to both sides of the equation.

$$x=\frac{2\sqrt{354}+36}{5} x=\frac{36-2\sqrt{354}}{5}$$