Solve 4x+5x=2x^2

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

Solve for x

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

$$x=0$$

Steps Using Factoring

Combine 4x and 5x to get 9x.

$$9x=2x^{2}$$

Subtract 2x^{2} from both sides.

$$9x-2x^{2}=0$$

Factor out x.

$$x\left(9-2x\right)=0$$

To find equation solutions, solve x=0 and 9-2x=0.

$$x=0 x=\frac{9}{2}$$

Steps Using the Quadratic Formula

Combine 4x and 5x to get 9x.

$$9x=2x^{2}$$

Subtract 2x^{2} from both sides.

$$9x-2x^{2}=0$$

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.

$$-2x^{2}+9x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 9 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}}}{2\left(-2\right)}$$

Take the square root of 9^{2}.

$$x=\frac{-9±9}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-9±9}{-4}$$

Now solve the equation x=\frac{-9±9}{-4} when ± is plus. Add -9 to 9.

$$x=\frac{0}{-4}$$

Divide 0 by -4.

$$x=0$$

Now solve the equation x=\frac{-9±9}{-4} when ± is minus. Subtract 9 from -9.

$$x=-\frac{18}{-4}$$

Reduce the fraction \frac{-18}{-4} to lowest terms by extracting and canceling out 2.

$$x=\frac{9}{2}$$

The equation is now solved.

$$x=0 x=\frac{9}{2}$$

Steps for Completing the Square

Combine 4x and 5x to get 9x.

$$9x=2x^{2}$$

Subtract 2x^{2} from both sides.

$$9x-2x^{2}=0$$

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.

$$-2x^{2}+9x=0$$

Divide both sides by -2.

$$\frac{-2x^{2}+9x}{-2}=\frac{0}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{9}{-2}x=\frac{0}{-2}$$

Divide 9 by -2.

$$x^{2}-\frac{9}{2}x=\frac{0}{-2}$$

Divide 0 by -2.

$$x^{2}-\frac{9}{2}x=0$$

Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=\left(-\frac{9}{4}\right)^{2}$$

Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{81}{16}$$

Factor x^{2}-\frac{9}{2}x+\frac{81}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{81}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Simplify.

$$x-\frac{9}{4}=\frac{9}{4} x-\frac{9}{4}=-\frac{9}{4}$$

Add \frac{9}{4} to both sides of the equation.

$$x=\frac{9}{2} x=0$$