Solve 36x=0.7x^2

$$36x=0.7 { x }^{ 2 }$$

Solve for x

$$x = \frac{360}{7} = 51\frac{3}{7} \approx 51.428571429$$

$$x=0$$

Steps Using Factoring

Subtract 0.7x^{2} from both sides.

$$36x-0.7x^{2}=0$$

Factor out x.

$$x\left(36-0.7x\right)=0$$

To find equation solutions, solve x=0 and 36-\frac{7x}{10}=0.

$$x=0 x=\frac{360}{7}$$

Steps Using the Quadratic Formula

Subtract 0.7x^{2} from both sides.

$$36x-0.7x^{2}=0$$

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.

$$-0.7x^{2}+36x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.7 for a, 36 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-36±\sqrt{36^{2}}}{2\left(-0.7\right)}$$

Take the square root of 36^{2}.

$$x=\frac{-36±36}{2\left(-0.7\right)}$$

Multiply 2 times -0.7.

$$x=\frac{-36±36}{-1.4}$$

Now solve the equation x=\frac{-36±36}{-1.4} when ± is plus. Add -36 to 36.

$$x=\frac{0}{-1.4}$$

Divide 0 by -1.4 by multiplying 0 by the reciprocal of -1.4.

$$x=0$$

Now solve the equation x=\frac{-36±36}{-1.4} when ± is minus. Subtract 36 from -36.

$$x=-\frac{72}{-1.4}$$

Divide -72 by -1.4 by multiplying -72 by the reciprocal of -1.4.

$$x=\frac{360}{7}$$

The equation is now solved.

$$x=0 x=\frac{360}{7}$$

Steps for Completing the Square

Subtract 0.7x^{2} from both sides.

$$36x-0.7x^{2}=0$$

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.

$$-0.7x^{2}+36x=0$$

Divide both sides of the equation by -0.7, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.7x^{2}+36x}{-0.7}=\frac{0}{-0.7}$$

Dividing by -0.7 undoes the multiplication by -0.7.

$$x^{2}+\frac{36}{-0.7}x=\frac{0}{-0.7}$$

Divide 36 by -0.7 by multiplying 36 by the reciprocal of -0.7.

$$x^{2}-\frac{360}{7}x=\frac{0}{-0.7}$$

Divide 0 by -0.7 by multiplying 0 by the reciprocal of -0.7.

$$x^{2}-\frac{360}{7}x=0$$

Divide -\frac{360}{7}, the coefficient of the x term, by 2 to get -\frac{180}{7}. Then add the square of -\frac{180}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{360}{7}x+\left(-\frac{180}{7}\right)^{2}=\left(-\frac{180}{7}\right)^{2}$$

Square -\frac{180}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{360}{7}x+\frac{32400}{49}=\frac{32400}{49}$$

Factor x^{2}-\frac{360}{7}x+\frac{32400}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{180}{7}\right)^{2}=\frac{32400}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{180}{7}\right)^{2}}=\sqrt{\frac{32400}{49}}$$

Simplify.

$$x-\frac{180}{7}=\frac{180}{7} x-\frac{180}{7}=-\frac{180}{7}$$

Add \frac{180}{7} to both sides of the equation.

$$x=\frac{360}{7} x=0$$