Solve (3x-78)(x+25)=(2x-9)(3x-78)

$$( 3 x - 78 ) ( x + 25 ) = ( 2 x - 9 ) ( 3 x - 78 )$$

Solve for x

$$x=34$$

$$x=26$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x-78 by x+25 and combine like terms.

$$3x^{2}-3x-1950=\left(2x-9\right)\left(3x-78\right)$$

Use the distributive property to multiply 2x-9 by 3x-78 and combine like terms.

$$3x^{2}-3x-1950=6x^{2}-183x+702$$

Subtract 6x^{2} from both sides.

$$3x^{2}-3x-1950-6x^{2}=-183x+702$$

Combine 3x^{2} and -6x^{2} to get -3x^{2}.

$$-3x^{2}-3x-1950=-183x+702$$

Add 183x to both sides.

$$-3x^{2}-3x-1950+183x=702$$

Combine -3x and 183x to get 180x.

$$-3x^{2}+180x-1950=702$$

Subtract 702 from both sides.

$$-3x^{2}+180x-1950-702=0$$

Subtract 702 from -1950 to get -2652.

$$-3x^{2}+180x-2652=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 180 for b, and -2652 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-180±\sqrt{180^{2}-4\left(-3\right)\left(-2652\right)}}{2\left(-3\right)}$$

Square 180.

$$x=\frac{-180±\sqrt{32400-4\left(-3\right)\left(-2652\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-180±\sqrt{32400+12\left(-2652\right)}}{2\left(-3\right)}$$

Multiply 12 times -2652.

$$x=\frac{-180±\sqrt{32400-31824}}{2\left(-3\right)}$$

Add 32400 to -31824.

$$x=\frac{-180±\sqrt{576}}{2\left(-3\right)}$$

Take the square root of 576.

$$x=\frac{-180±24}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-180±24}{-6}$$

Now solve the equation x=\frac{-180±24}{-6} when ± is plus. Add -180 to 24.

$$x=-\frac{156}{-6}$$

Divide -156 by -6.

$$x=26$$

Now solve the equation x=\frac{-180±24}{-6} when ± is minus. Subtract 24 from -180.

$$x=-\frac{204}{-6}$$

Divide -204 by -6.

$$x=34$$

The equation is now solved.

$$x=26 x=34$$

Steps for Completing the Square

Use the distributive property to multiply 3x-78 by x+25 and combine like terms.

$$3x^{2}-3x-1950=\left(2x-9\right)\left(3x-78\right)$$

Use the distributive property to multiply 2x-9 by 3x-78 and combine like terms.

$$3x^{2}-3x-1950=6x^{2}-183x+702$$

Subtract 6x^{2} from both sides.

$$3x^{2}-3x-1950-6x^{2}=-183x+702$$

Combine 3x^{2} and -6x^{2} to get -3x^{2}.

$$-3x^{2}-3x-1950=-183x+702$$

Add 183x to both sides.

$$-3x^{2}-3x-1950+183x=702$$

Combine -3x and 183x to get 180x.

$$-3x^{2}+180x-1950=702$$

Add 1950 to both sides.

$$-3x^{2}+180x=702+1950$$

Add 702 and 1950 to get 2652.

$$-3x^{2}+180x=2652$$

Divide both sides by -3.

$$\frac{-3x^{2}+180x}{-3}=\frac{2652}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{180}{-3}x=\frac{2652}{-3}$$

Divide 180 by -3.

$$x^{2}-60x=\frac{2652}{-3}$$

Divide 2652 by -3.

$$x^{2}-60x=-884$$

Divide -60, the coefficient of the x term, by 2 to get -30. Then add the square of -30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-60x+\left(-30\right)^{2}=-884+\left(-30\right)^{2}$$

Square -30.

$$x^{2}-60x+900=-884+900$$

Add -884 to 900.

$$x^{2}-60x+900=16$$

Factor x^{2}-60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-30\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-30\right)^{2}}=\sqrt{16}$$

Simplify.

$$x-30=4 x-30=-4$$

Add 30 to both sides of the equation.

$$x=34 x=26$$