Solve 3*left(2x-10right)*2left(3x-30right)=-5*left(3x+100right)

$$3 \times \left( 2x-10 \right) \times 2 \left( 3x-30 \right) = -5 \times \left( 3x+100 \right)$$

Solve for x (complex solution)

$$x=\frac{175+5\sqrt{247}i}{24}\approx 7.291666667+3.274215343i$$

$$x=\frac{-5\sqrt{247}i+175}{24}\approx 7.291666667-3.274215343i$$

Steps Using the Quadratic Formula

Multiply 3 and 2 to get 6.

$$6\left(2x-10\right)\left(3x-30\right)=-5\left(3x+100\right)$$

Use the distributive property to multiply 6 by 2x-10.

$$\left(12x-60\right)\left(3x-30\right)=-5\left(3x+100\right)$$

Use the distributive property to multiply 12x-60 by 3x-30 and combine like terms.

$$36x^{2}-540x+1800=-5\left(3x+100\right)$$

Use the distributive property to multiply -5 by 3x+100.

$$36x^{2}-540x+1800=-15x-500$$

Add 15x to both sides.

$$36x^{2}-540x+1800+15x=-500$$

Combine -540x and 15x to get -525x.

$$36x^{2}-525x+1800=-500$$

Add 500 to both sides.

$$36x^{2}-525x+1800+500=0$$

Add 1800 and 500 to get 2300.

$$36x^{2}-525x+2300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, -525 for b, and 2300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-525\right)±\sqrt{\left(-525\right)^{2}-4\times 36\times 2300}}{2\times 36}$$

Square -525.

$$x=\frac{-\left(-525\right)±\sqrt{275625-4\times 36\times 2300}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-\left(-525\right)±\sqrt{275625-144\times 2300}}{2\times 36}$$

Multiply -144 times 2300.

$$x=\frac{-\left(-525\right)±\sqrt{275625-331200}}{2\times 36}$$

Add 275625 to -331200.

$$x=\frac{-\left(-525\right)±\sqrt{-55575}}{2\times 36}$$

Take the square root of -55575.

$$x=\frac{-\left(-525\right)±15\sqrt{247}i}{2\times 36}$$

The opposite of -525 is 525.

$$x=\frac{525±15\sqrt{247}i}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{525±15\sqrt{247}i}{72}$$

Now solve the equation x=\frac{525±15\sqrt{247}i}{72} when ± is plus. Add 525 to 15i\sqrt{247}.

$$x=\frac{525+15\sqrt{247}i}{72}$$

Divide 525+15i\sqrt{247} by 72.

$$x=\frac{175+5\sqrt{247}i}{24}$$

Now solve the equation x=\frac{525±15\sqrt{247}i}{72} when ± is minus. Subtract 15i\sqrt{247} from 525.

$$x=\frac{-15\sqrt{247}i+525}{72}$$

Divide 525-15i\sqrt{247} by 72.

$$x=\frac{-5\sqrt{247}i+175}{24}$$

The equation is now solved.

$$x=\frac{175+5\sqrt{247}i}{24} x=\frac{-5\sqrt{247}i+175}{24}$$

Steps for Completing the Square

Multiply 3 and 2 to get 6.

$$6\left(2x-10\right)\left(3x-30\right)=-5\left(3x+100\right)$$

Use the distributive property to multiply 6 by 2x-10.

$$\left(12x-60\right)\left(3x-30\right)=-5\left(3x+100\right)$$

Use the distributive property to multiply 12x-60 by 3x-30 and combine like terms.

$$36x^{2}-540x+1800=-5\left(3x+100\right)$$

Use the distributive property to multiply -5 by 3x+100.

$$36x^{2}-540x+1800=-15x-500$$

Add 15x to both sides.

$$36x^{2}-540x+1800+15x=-500$$

Combine -540x and 15x to get -525x.

$$36x^{2}-525x+1800=-500$$

Subtract 1800 from both sides.

$$36x^{2}-525x=-500-1800$$

Subtract 1800 from -500 to get -2300.

$$36x^{2}-525x=-2300$$

Divide both sides by 36.

$$\frac{36x^{2}-525x}{36}=-\frac{2300}{36}$$

Dividing by 36 undoes the multiplication by 36.

$$x^{2}+\left(-\frac{525}{36}\right)x=-\frac{2300}{36}$$

Reduce the fraction \frac{-525}{36} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{175}{12}x=-\frac{2300}{36}$$

Reduce the fraction \frac{-2300}{36} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{175}{12}x=-\frac{575}{9}$$

Divide -\frac{175}{12}, the coefficient of the x term, by 2 to get -\frac{175}{24}. Then add the square of -\frac{175}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{175}{12}x+\left(-\frac{175}{24}\right)^{2}=-\frac{575}{9}+\left(-\frac{175}{24}\right)^{2}$$

Square -\frac{175}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{175}{12}x+\frac{30625}{576}=-\frac{575}{9}+\frac{30625}{576}$$

Add -\frac{575}{9} to \frac{30625}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{175}{12}x+\frac{30625}{576}=-\frac{6175}{576}$$

Factor x^{2}-\frac{175}{12}x+\frac{30625}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{175}{24}\right)^{2}=-\frac{6175}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{175}{24}\right)^{2}}=\sqrt{-\frac{6175}{576}}$$

Simplify.

$$x-\frac{175}{24}=\frac{5\sqrt{247}i}{24} x-\frac{175}{24}=-\frac{5\sqrt{247}i}{24}$$

Add \frac{175}{24} to both sides of the equation.

$$x=\frac{175+5\sqrt{247}i}{24} x=\frac{-5\sqrt{247}i+175}{24}$$