Solve (x+5)(x-8)=2x(x+5)+3x(x-8)

$$( x + 5 ) ( x - 8 ) = 2 x ( x + 5 ) + 3 x ( x - 8 )$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{519}i+11}{8}\approx 1.375-2.847696437i$$

$$x=\frac{11+\sqrt{519}i}{8}\approx 1.375+2.847696437i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+5 by x-8 and combine like terms.

$$x^{2}-3x-40=2x\left(x+5\right)+3x\left(x-8\right)$$

Use the distributive property to multiply 2x by x+5.

$$x^{2}-3x-40=2x^{2}+10x+3x\left(x-8\right)$$

Use the distributive property to multiply 3x by x-8.

$$x^{2}-3x-40=2x^{2}+10x+3x^{2}-24x$$

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

$$x^{2}-3x-40=5x^{2}+10x-24x$$

Combine 10x and -24x to get -14x.

$$x^{2}-3x-40=5x^{2}-14x$$

Subtract 5x^{2} from both sides.

$$x^{2}-3x-40-5x^{2}=-14x$$

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}-3x-40=-14x$$

Add 14x to both sides.

$$-4x^{2}-3x-40+14x=0$$

Combine -3x and 14x to get 11x.

$$-4x^{2}+11x-40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 11 for b, and -40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-4\right)\left(-40\right)}}{2\left(-4\right)}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\left(-4\right)\left(-40\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-11±\sqrt{121+16\left(-40\right)}}{2\left(-4\right)}$$

Multiply 16 times -40.

$$x=\frac{-11±\sqrt{121-640}}{2\left(-4\right)}$$

Add 121 to -640.

$$x=\frac{-11±\sqrt{-519}}{2\left(-4\right)}$$

Take the square root of -519.

$$x=\frac{-11±\sqrt{519}i}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-11±\sqrt{519}i}{-8}$$

Now solve the equation x=\frac{-11±\sqrt{519}i}{-8} when ± is plus. Add -11 to i\sqrt{519}.

$$x=\frac{-11+\sqrt{519}i}{-8}$$

Divide -11+i\sqrt{519} by -8.

$$x=\frac{-\sqrt{519}i+11}{8}$$

Now solve the equation x=\frac{-11±\sqrt{519}i}{-8} when ± is minus. Subtract i\sqrt{519} from -11.

$$x=\frac{-\sqrt{519}i-11}{-8}$$

Divide -11-i\sqrt{519} by -8.

$$x=\frac{11+\sqrt{519}i}{8}$$

The equation is now solved.

$$x=\frac{-\sqrt{519}i+11}{8} x=\frac{11+\sqrt{519}i}{8}$$

Steps for Completing the Square

Use the distributive property to multiply x+5 by x-8 and combine like terms.

$$x^{2}-3x-40=2x\left(x+5\right)+3x\left(x-8\right)$$

Use the distributive property to multiply 2x by x+5.

$$x^{2}-3x-40=2x^{2}+10x+3x\left(x-8\right)$$

Use the distributive property to multiply 3x by x-8.

$$x^{2}-3x-40=2x^{2}+10x+3x^{2}-24x$$

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

$$x^{2}-3x-40=5x^{2}+10x-24x$$

Combine 10x and -24x to get -14x.

$$x^{2}-3x-40=5x^{2}-14x$$

Subtract 5x^{2} from both sides.

$$x^{2}-3x-40-5x^{2}=-14x$$

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}-3x-40=-14x$$

Add 14x to both sides.

$$-4x^{2}-3x-40+14x=0$$

Combine -3x and 14x to get 11x.

$$-4x^{2}+11x-40=0$$

Add 40 to both sides. Anything plus zero gives itself.

$$-4x^{2}+11x=40$$

Divide both sides by -4.

$$\frac{-4x^{2}+11x}{-4}=\frac{40}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{11}{-4}x=\frac{40}{-4}$$

Divide 11 by -4.

$$x^{2}-\frac{11}{4}x=\frac{40}{-4}$$

Divide 40 by -4.

$$x^{2}-\frac{11}{4}x=-10$$

Divide -\frac{11}{4}, the coefficient of the x term, by 2 to get -\frac{11}{8}. Then add the square of -\frac{11}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{4}x+\left(-\frac{11}{8}\right)^{2}=-10+\left(-\frac{11}{8}\right)^{2}$$

Square -\frac{11}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=-10+\frac{121}{64}$$

Add -10 to \frac{121}{64}.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=-\frac{519}{64}$$

Factor x^{2}-\frac{11}{4}x+\frac{121}{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}.

$$\left(x-\frac{11}{8}\right)^{2}=-\frac{519}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{8}\right)^{2}}=\sqrt{-\frac{519}{64}}$$

Simplify.

$$x-\frac{11}{8}=\frac{\sqrt{519}i}{8} x-\frac{11}{8}=-\frac{\sqrt{519}i}{8}$$

Add \frac{11}{8} to both sides of the equation.

$$x=\frac{11+\sqrt{519}i}{8} x=\frac{-\sqrt{519}i+11}{8}$$