Solve 3x(x+1)=5x(3-x)

$$3 x ( x + 1 ) = 5 x ( 3 - x )$$

Solve for x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply 3x by x+1.

$$3x^{2}+3x=5x\left(3-x\right)$$

Use the distributive property to multiply 5x by 3-x.

$$3x^{2}+3x=15x-5x^{2}$$

Subtract 15x from both sides.

$$3x^{2}+3x-15x=-5x^{2}$$

Combine 3x and -15x to get -12x.

$$3x^{2}-12x=-5x^{2}$$

Add 5x^{2} to both sides.

$$3x^{2}-12x+5x^{2}=0$$

Combine 3x^{2} and 5x^{2} to get 8x^{2}.

$$8x^{2}-12x=0$$

Factor out x.

$$x\left(8x-12\right)=0$$

To find equation solutions, solve x=0 and 8x-12=0.

$$x=0 x=\frac{3}{2}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x by x+1.

$$3x^{2}+3x=5x\left(3-x\right)$$

Use the distributive property to multiply 5x by 3-x.

$$3x^{2}+3x=15x-5x^{2}$$

Subtract 15x from both sides.

$$3x^{2}+3x-15x=-5x^{2}$$

Combine 3x and -15x to get -12x.

$$3x^{2}-12x=-5x^{2}$$

Add 5x^{2} to both sides.

$$3x^{2}-12x+5x^{2}=0$$

Combine 3x^{2} and 5x^{2} to get 8x^{2}.

$$8x^{2}-12x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -12 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}}}{2\times 8}$$

Take the square root of \left(-12\right)^{2}.

$$x=\frac{-\left(-12\right)±12}{2\times 8}$$

The opposite of -12 is 12.

$$x=\frac{12±12}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{12±12}{16}$$

Now solve the equation x=\frac{12±12}{16} when ± is plus. Add 12 to 12.

$$x=\frac{24}{16}$$

Reduce the fraction \frac{24}{16} to lowest terms by extracting and canceling out 8.

$$x=\frac{3}{2}$$

Now solve the equation x=\frac{12±12}{16} when ± is minus. Subtract 12 from 12.

$$x=\frac{0}{16}$$

Divide 0 by 16.

$$x=0$$

The equation is now solved.

$$x=\frac{3}{2} x=0$$

Steps for Completing the Square

Use the distributive property to multiply 3x by x+1.

$$3x^{2}+3x=5x\left(3-x\right)$$

Use the distributive property to multiply 5x by 3-x.

$$3x^{2}+3x=15x-5x^{2}$$

Subtract 15x from both sides.

$$3x^{2}+3x-15x=-5x^{2}$$

Combine 3x and -15x to get -12x.

$$3x^{2}-12x=-5x^{2}$$

Add 5x^{2} to both sides.

$$3x^{2}-12x+5x^{2}=0$$

Combine 3x^{2} and 5x^{2} to get 8x^{2}.

$$8x^{2}-12x=0$$

Divide both sides by 8.

$$\frac{8x^{2}-12x}{8}=\frac{0}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\left(-\frac{12}{8}\right)x=\frac{0}{8}$$

Reduce the fraction \frac{-12}{8} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{3}{2}x=\frac{0}{8}$$

Divide 0 by 8.

$$x^{2}-\frac{3}{2}x=0$$

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{9}{16}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{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}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{9}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Simplify.

$$x-\frac{3}{4}=\frac{3}{4} x-\frac{3}{4}=-\frac{3}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$x=\frac{3}{2} x=0$$