Solve 6x+5*1=-4*5-5(sqrt{x+4}+5

$$6 x + 5 \cdot 1 = - 4 \cdot 5 - 5 ( \sqrt { x + 4 } + 5$$

Solve for x (complex solution)

$$x=\frac{-5\sqrt{599}i-575}{72}\approx -7.986111111-1.699616424i$$

Solution Steps

Multiply 5 and 1 to get 5.

$$6x+5=-4\times 5-5\left(\sqrt{x+4}+5\right)$$

Multiply -4 and 5 to get -20.

$$6x+5=-20-5\left(\sqrt{x+4}+5\right)$$

Add 5\left(\sqrt{x+4}+5\right) to both sides.

$$6x+5+5\left(\sqrt{x+4}+5\right)=-20$$

Use the distributive property to multiply 5 by \sqrt{x+4}+5.

$$6x+5+5\sqrt{x+4}+25=-20$$

Add 5 and 25 to get 30.

$$6x+30+5\sqrt{x+4}=-20$$

Subtract 30 from both sides.

$$6x+5\sqrt{x+4}=-20-30$$

Subtract 30 from -20 to get -50.

$$6x+5\sqrt{x+4}=-50$$

Subtract 6x from both sides of the equation.

$$5\sqrt{x+4}=-50-6x$$

Square both sides of the equation.

$$\left(5\sqrt{x+4}\right)^{2}=\left(-50-6x\right)^{2}$$

Expand \left(5\sqrt{x+4}\right)^{2}.

$$5^{2}\left(\sqrt{x+4}\right)^{2}=\left(-50-6x\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25\left(\sqrt{x+4}\right)^{2}=\left(-50-6x\right)^{2}$$

Calculate \sqrt{x+4} to the power of 2 and get x+4.

$$25\left(x+4\right)=\left(-50-6x\right)^{2}$$

Use the distributive property to multiply 25 by x+4.

$$25x+100=\left(-50-6x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-50-6x\right)^{2}.

$$25x+100=2500+600x+36x^{2}$$

Subtract 600x from both sides.

$$25x+100-600x=2500+36x^{2}$$

Combine 25x and -600x to get -575x.

$$-575x+100=2500+36x^{2}$$

Subtract 36x^{2} from both sides.

$$-575x+100-36x^{2}=2500$$

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.

$$-36x^{2}-575x+100=2500$$

Subtract 2500 from both sides of the equation.

$$-36x^{2}-575x+100-2500=2500-2500$$

Subtracting 2500 from itself leaves 0.

$$-36x^{2}-575x+100-2500=0$$

Subtract 2500 from 100.

$$-36x^{2}-575x-2400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -36 for a, -575 for b, and -2400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-575\right)±\sqrt{\left(-575\right)^{2}-4\left(-36\right)\left(-2400\right)}}{2\left(-36\right)}$$

Square -575.

$$x=\frac{-\left(-575\right)±\sqrt{330625-4\left(-36\right)\left(-2400\right)}}{2\left(-36\right)}$$

Multiply -4 times -36.

$$x=\frac{-\left(-575\right)±\sqrt{330625+144\left(-2400\right)}}{2\left(-36\right)}$$

Multiply 144 times -2400.

$$x=\frac{-\left(-575\right)±\sqrt{330625-345600}}{2\left(-36\right)}$$

Add 330625 to -345600.

$$x=\frac{-\left(-575\right)±\sqrt{-14975}}{2\left(-36\right)}$$

Take the square root of -14975.

$$x=\frac{-\left(-575\right)±5\sqrt{599}i}{2\left(-36\right)}$$

The opposite of -575 is 575.

$$x=\frac{575±5\sqrt{599}i}{2\left(-36\right)}$$

Multiply 2 times -36.

$$x=\frac{575±5\sqrt{599}i}{-72}$$

Now solve the equation x=\frac{575±5\sqrt{599}i}{-72} when ± is plus. Add 575 to 5i\sqrt{599}.

$$x=\frac{575+5\sqrt{599}i}{-72}$$

Divide 575+5i\sqrt{599} by -72.

$$x=\frac{-5\sqrt{599}i-575}{72}$$

Now solve the equation x=\frac{575±5\sqrt{599}i}{-72} when ± is minus. Subtract 5i\sqrt{599} from 575.

$$x=\frac{-5\sqrt{599}i+575}{-72}$$

Divide 575-5i\sqrt{599} by -72.

$$x=\frac{-575+5\sqrt{599}i}{72}$$

The equation is now solved.

$$x=\frac{-5\sqrt{599}i-575}{72} x=\frac{-575+5\sqrt{599}i}{72}$$

Substitute \frac{-5\sqrt{599}i-575}{72} for x in the equation 6x+5\times 1=-4\times 5-5\left(\sqrt{x+4}+5\right).

$$6\times \frac{-5\sqrt{599}i-575}{72}+5\times 1=-4\times 5-5\left(\sqrt{\frac{-5\sqrt{599}i-575}{72}+4}+5\right)$$

Simplify. The value x=\frac{-5\sqrt{599}i-575}{72} satisfies the equation.

$$-\frac{5}{12}i\times 599^{\frac{1}{2}}-\frac{515}{12}=-\frac{515}{12}-\frac{5}{12}i\times 599^{\frac{1}{2}}$$

Substitute \frac{-575+5\sqrt{599}i}{72} for x in the equation 6x+5\times 1=-4\times 5-5\left(\sqrt{x+4}+5\right).

$$6\times \frac{-575+5\sqrt{599}i}{72}+5\times 1=-4\times 5-5\left(\sqrt{\frac{-575+5\sqrt{599}i}{72}+4}+5\right)$$

Simplify. The value x=\frac{-575+5\sqrt{599}i}{72} does not satisfy the equation.

$$-\frac{515}{12}+\frac{5}{12}i\times 599^{\frac{1}{2}}=-\frac{565}{12}-\frac{5}{12}i\times 599^{\frac{1}{2}}$$

Equation 5\sqrt{x+4}=-6x-50 has a unique solution.

$$x=\frac{-5\sqrt{599}i-575}{72}$$