Solve 3x(x-4)=(2-x)(x+5

$$3 x ( x - 4 ) = ( 2 - x ) ( x + 5$$

Solve for x

$$x = \frac{\sqrt{241} + 9}{8} \approx 3.065521837$$

$$x=\frac{9-\sqrt{241}}{8}\approx -0.815521837$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x by x-4.

$$3x^{2}-12x=\left(2-x\right)\left(x+5\right)$$

Use the distributive property to multiply 2-x by x+5 and combine like terms.

$$3x^{2}-12x=-3x+10-x^{2}$$

Add 3x to both sides.

$$3x^{2}-12x+3x=10-x^{2}$$

Combine -12x and 3x to get -9x.

$$3x^{2}-9x=10-x^{2}$$

Subtract 10 from both sides.

$$3x^{2}-9x-10=-x^{2}$$

Add x^{2} to both sides.

$$3x^{2}-9x-10+x^{2}=0$$

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$4x^{2}-9x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -9 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 4\left(-10\right)}}{2\times 4}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 4\left(-10\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-9\right)±\sqrt{81-16\left(-10\right)}}{2\times 4}$$

Multiply -16 times -10.

$$x=\frac{-\left(-9\right)±\sqrt{81+160}}{2\times 4}$$

Add 81 to 160.

$$x=\frac{-\left(-9\right)±\sqrt{241}}{2\times 4}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{241}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{9±\sqrt{241}}{8}$$

Now solve the equation x=\frac{9±\sqrt{241}}{8} when ± is plus. Add 9 to \sqrt{241}.

$$x=\frac{\sqrt{241}+9}{8}$$

Now solve the equation x=\frac{9±\sqrt{241}}{8} when ± is minus. Subtract \sqrt{241} from 9.

$$x=\frac{9-\sqrt{241}}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{241}+9}{8} x=\frac{9-\sqrt{241}}{8}$$

Steps for Completing the Square

Use the distributive property to multiply 3x by x-4.

$$3x^{2}-12x=\left(2-x\right)\left(x+5\right)$$

Use the distributive property to multiply 2-x by x+5 and combine like terms.

$$3x^{2}-12x=-3x+10-x^{2}$$

Add 3x to both sides.

$$3x^{2}-12x+3x=10-x^{2}$$

Combine -12x and 3x to get -9x.

$$3x^{2}-9x=10-x^{2}$$

Add x^{2} to both sides.

$$3x^{2}-9x+x^{2}=10$$

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$4x^{2}-9x=10$$

Divide both sides by 4.

$$\frac{4x^{2}-9x}{4}=\frac{10}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{9}{4}x=\frac{10}{4}$$

Reduce the fraction \frac{10}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{9}{4}x=\frac{5}{2}$$

Divide -\frac{9}{4}, the coefficient of the x term, by 2 to get -\frac{9}{8}. Then add the square of -\frac{9}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{4}x+\left(-\frac{9}{8}\right)^{2}=\frac{5}{2}+\left(-\frac{9}{8}\right)^{2}$$

Square -\frac{9}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=\frac{5}{2}+\frac{81}{64}$$

Add \frac{5}{2} to \frac{81}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=\frac{241}{64}$$

Factor x^{2}-\frac{9}{4}x+\frac{81}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{8}\right)^{2}=\frac{241}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{8}\right)^{2}}=\sqrt{\frac{241}{64}}$$

Simplify.

$$x-\frac{9}{8}=\frac{\sqrt{241}}{8} x-\frac{9}{8}=-\frac{\sqrt{241}}{8}$$

Add \frac{9}{8} to both sides of the equation.

$$x=\frac{\sqrt{241}+9}{8} x=\frac{9-\sqrt{241}}{8}$$