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

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

Solve for x

$$x=-\frac{1}{5}=-0.2$$

$$x=0$$

Steps Using Factoring

Combine 5x and -2x to get 3x.

$$4x+5x^{2}=3x$$

Subtract 3x from both sides.

$$4x+5x^{2}-3x=0$$

Combine 4x and -3x to get x.

$$x+5x^{2}=0$$

Factor out x.

$$x\left(1+5x\right)=0$$

To find equation solutions, solve x=0 and 1+5x=0.

$$x=0 x=-\frac{1}{5}$$

Steps Using the Quadratic Formula

Combine 5x and -2x to get 3x.

$$4x+5x^{2}=3x$$

Subtract 3x from both sides.

$$4x+5x^{2}-3x=0$$

Combine 4x and -3x to get x.

$$x+5x^{2}=0$$

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.

$$5x^{2}+x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 1 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}}}{2\times 5}$$

Take the square root of 1^{2}.

$$x=\frac{-1±1}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-1±1}{10}$$

Now solve the equation x=\frac{-1±1}{10} when ± is plus. Add -1 to 1.

$$x=\frac{0}{10}$$

Divide 0 by 10.

$$x=0$$

Now solve the equation x=\frac{-1±1}{10} when ± is minus. Subtract 1 from -1.

$$x=-\frac{2}{10}$$

Reduce the fraction \frac{-2}{10} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{5}$$

The equation is now solved.

$$x=0 x=-\frac{1}{5}$$

Steps for Completing the Square

Combine 5x and -2x to get 3x.

$$4x+5x^{2}=3x$$

Subtract 3x from both sides.

$$4x+5x^{2}-3x=0$$

Combine 4x and -3x to get x.

$$x+5x^{2}=0$$

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.

$$5x^{2}+x=0$$

Divide both sides by 5.

$$\frac{5x^{2}+x}{5}=\frac{0}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{1}{5}x=\frac{0}{5}$$

Divide 0 by 5.

$$x^{2}+\frac{1}{5}x=0$$

Divide \frac{1}{5}, the coefficient of the x term, by 2 to get \frac{1}{10}. Then add the square of \frac{1}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{5}x+\left(\frac{1}{10}\right)^{2}=\left(\frac{1}{10}\right)^{2}$$

Square \frac{1}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{5}x+\frac{1}{100}=\frac{1}{100}$$

Factor x^{2}+\frac{1}{5}x+\frac{1}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{10}\right)^{2}=\frac{1}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{10}\right)^{2}}=\sqrt{\frac{1}{100}}$$

Simplify.

$$x+\frac{1}{10}=\frac{1}{10} x+\frac{1}{10}=-\frac{1}{10}$$

Subtract \frac{1}{10} from both sides of the equation.

$$x=0 x=-\frac{1}{5}$$