Solve 3sqrt{4(3x-5)}+2(7x+3)7=16x-3(4x-8)

$$3 \sqrt { 4 ( 3 x - 5 ) } + 2 ( 7 x + 3 ) 7 = 16 x - 3 ( 4 x - 8 )$$

Solve for x (complex solution)

$$x=\frac{-15\sqrt{1967}i-819}{4418}\approx -0.185377999-0.150580151i$$

Solution Steps

Subtract 2\left(7x+3\right)\times 7 from both sides of the equation.

$$3\sqrt{4\left(3x-5\right)}=16x-3\left(4x-8\right)-2\left(7x+3\right)\times 7$$

Use the distributive property to multiply 4 by 3x-5.

$$3\sqrt{12x-20}=16x-3\left(4x-8\right)-2\left(7x+3\right)\times 7$$

Use the distributive property to multiply -3 by 4x-8.

$$3\sqrt{12x-20}=16x-12x+24-2\left(7x+3\right)\times 7$$

Combine 16x and -12x to get 4x.

$$3\sqrt{12x-20}=4x+24-2\left(7x+3\right)\times 7$$

Multiply -2 and 7 to get -14.

$$3\sqrt{12x-20}=4x+24-14\left(7x+3\right)$$

Use the distributive property to multiply -14 by 7x+3.

$$3\sqrt{12x-20}=4x+24-98x-42$$

Combine 4x and -98x to get -94x.

$$3\sqrt{12x-20}=-94x+24-42$$

Subtract 42 from 24 to get -18.

$$3\sqrt{12x-20}=-94x-18$$

Square both sides of the equation.

$$\left(3\sqrt{12x-20}\right)^{2}=\left(-94x-18\right)^{2}$$

Expand \left(3\sqrt{12x-20}\right)^{2}.

$$3^{2}\left(\sqrt{12x-20}\right)^{2}=\left(-94x-18\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{12x-20}\right)^{2}=\left(-94x-18\right)^{2}$$

Calculate \sqrt{12x-20} to the power of 2 and get 12x-20.

$$9\left(12x-20\right)=\left(-94x-18\right)^{2}$$

Use the distributive property to multiply 9 by 12x-20.

$$108x-180=\left(-94x-18\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-94x-18\right)^{2}.

$$108x-180=8836x^{2}+3384x+324$$

Subtract 8836x^{2} from both sides.

$$108x-180-8836x^{2}=3384x+324$$

Subtract 3384x from both sides.

$$108x-180-8836x^{2}-3384x=324$$

Combine 108x and -3384x to get -3276x.

$$-3276x-180-8836x^{2}=324$$

Subtract 324 from both sides.

$$-3276x-180-8836x^{2}-324=0$$

Subtract 324 from -180 to get -504.

$$-3276x-504-8836x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-8836x^{2}-3276x-504=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8836 for a, -3276 for b, and -504 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3276\right)±\sqrt{\left(-3276\right)^{2}-4\left(-8836\right)\left(-504\right)}}{2\left(-8836\right)}$$

Square -3276.

$$x=\frac{-\left(-3276\right)±\sqrt{10732176-4\left(-8836\right)\left(-504\right)}}{2\left(-8836\right)}$$

Multiply -4 times -8836.

$$x=\frac{-\left(-3276\right)±\sqrt{10732176+35344\left(-504\right)}}{2\left(-8836\right)}$$

Multiply 35344 times -504.

$$x=\frac{-\left(-3276\right)±\sqrt{10732176-17813376}}{2\left(-8836\right)}$$

Add 10732176 to -17813376.

$$x=\frac{-\left(-3276\right)±\sqrt{-7081200}}{2\left(-8836\right)}$$

Take the square root of -7081200.

$$x=\frac{-\left(-3276\right)±60\sqrt{1967}i}{2\left(-8836\right)}$$

The opposite of -3276 is 3276.

$$x=\frac{3276±60\sqrt{1967}i}{2\left(-8836\right)}$$

Multiply 2 times -8836.

$$x=\frac{3276±60\sqrt{1967}i}{-17672}$$

Now solve the equation x=\frac{3276±60\sqrt{1967}i}{-17672} when ± is plus. Add 3276 to 60i\sqrt{1967}.

$$x=\frac{3276+60\sqrt{1967}i}{-17672}$$

Divide 3276+60i\sqrt{1967} by -17672.

$$x=\frac{-15\sqrt{1967}i-819}{4418}$$

Now solve the equation x=\frac{3276±60\sqrt{1967}i}{-17672} when ± is minus. Subtract 60i\sqrt{1967} from 3276.

$$x=\frac{-60\sqrt{1967}i+3276}{-17672}$$

Divide 3276-60i\sqrt{1967} by -17672.

$$x=\frac{-819+15\sqrt{1967}i}{4418}$$

The equation is now solved.

$$x=\frac{-15\sqrt{1967}i-819}{4418} x=\frac{-819+15\sqrt{1967}i}{4418}$$

Substitute \frac{-15\sqrt{1967}i-819}{4418} for x in the equation 3\sqrt{4\left(3x-5\right)}+2\left(7x+3\right)\times 7=16x-3\left(4x-8\right).

$$3\sqrt{4\left(3\times \frac{-15\sqrt{1967}i-819}{4418}-5\right)}+2\left(7\times \frac{-15\sqrt{1967}i-819}{4418}+3\right)\times 7=16\times \frac{-15\sqrt{1967}i-819}{4418}-3\left(4\times \frac{-15\sqrt{1967}i-819}{4418}-8\right)$$

Simplify. The value x=\frac{-15\sqrt{1967}i-819}{4418} satisfies the equation.

$$\frac{51378}{2209}-\frac{30}{2209}i\times 1967^{\frac{1}{2}}=-\frac{30}{2209}i\times 1967^{\frac{1}{2}}+\frac{51378}{2209}$$

Substitute \frac{-819+15\sqrt{1967}i}{4418} for x in the equation 3\sqrt{4\left(3x-5\right)}+2\left(7x+3\right)\times 7=16x-3\left(4x-8\right).

$$3\sqrt{4\left(3\times \frac{-819+15\sqrt{1967}i}{4418}-5\right)}+2\left(7\times \frac{-819+15\sqrt{1967}i}{4418}+3\right)\times 7=16\times \frac{-819+15\sqrt{1967}i}{4418}-3\left(4\times \frac{-819+15\sqrt{1967}i}{4418}-8\right)$$

Simplify. The value x=\frac{-819+15\sqrt{1967}i}{4418} does not satisfy the equation.

$$\frac{53916}{2209}+\frac{1440}{2209}i\times 1967^{\frac{1}{2}}=\frac{51378}{2209}+\frac{30}{2209}i\times 1967^{\frac{1}{2}}$$

Equation 3\sqrt{12x-20}=-94x-18 has a unique solution.

$$x=\frac{-15\sqrt{1967}i-819}{4418}$$