Solve -2sqrt{6x^2-34+48}=-13-5x

$$-2 \sqrt{ 6 { x }^{ 2 } -34+48 } = -13-5x$$

Solve for x

$$x=4\sqrt{257}-65\approx -0.875121832$$

Solution Steps

Square both sides of the equation.

$$\left(-2\sqrt{6x^{2}-34+48}\right)^{2}=\left(-13-5x\right)^{2}$$

Add -34 and 48 to get 14.

$$\left(-2\sqrt{6x^{2}+14}\right)^{2}=\left(-13-5x\right)^{2}$$

Expand \left(-2\sqrt{6x^{2}+14}\right)^{2}.

$$\left(-2\right)^{2}\left(\sqrt{6x^{2}+14}\right)^{2}=\left(-13-5x\right)^{2}$$

Calculate -2 to the power of 2 and get 4.

$$4\left(\sqrt{6x^{2}+14}\right)^{2}=\left(-13-5x\right)^{2}$$

Calculate \sqrt{6x^{2}+14} to the power of 2 and get 6x^{2}+14.

$$4\left(6x^{2}+14\right)=\left(-13-5x\right)^{2}$$

Use the distributive property to multiply 4 by 6x^{2}+14.

$$24x^{2}+56=\left(-13-5x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-13-5x\right)^{2}.

$$24x^{2}+56=169+130x+25x^{2}$$

Subtract 169 from both sides.

$$24x^{2}+56-169=130x+25x^{2}$$

Subtract 169 from 56 to get -113.

$$24x^{2}-113=130x+25x^{2}$$

Subtract 130x from both sides.

$$24x^{2}-113-130x=25x^{2}$$

Subtract 25x^{2} from both sides.

$$24x^{2}-113-130x-25x^{2}=0$$

Combine 24x^{2} and -25x^{2} to get -x^{2}.

$$-x^{2}-113-130x=0$$

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.

$$-x^{2}-130x-113=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -130 for b, and -113 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-130\right)±\sqrt{\left(-130\right)^{2}-4\left(-1\right)\left(-113\right)}}{2\left(-1\right)}$$

Square -130.

$$x=\frac{-\left(-130\right)±\sqrt{16900-4\left(-1\right)\left(-113\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-130\right)±\sqrt{16900+4\left(-113\right)}}{2\left(-1\right)}$$

Multiply 4 times -113.

$$x=\frac{-\left(-130\right)±\sqrt{16900-452}}{2\left(-1\right)}$$

Add 16900 to -452.

$$x=\frac{-\left(-130\right)±\sqrt{16448}}{2\left(-1\right)}$$

Take the square root of 16448.

$$x=\frac{-\left(-130\right)±8\sqrt{257}}{2\left(-1\right)}$$

The opposite of -130 is 130.

$$x=\frac{130±8\sqrt{257}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{130±8\sqrt{257}}{-2}$$

Now solve the equation x=\frac{130±8\sqrt{257}}{-2} when ± is plus. Add 130 to 8\sqrt{257}.

$$x=\frac{8\sqrt{257}+130}{-2}$$

Divide 130+8\sqrt{257} by -2.

$$x=-4\sqrt{257}-65$$

Now solve the equation x=\frac{130±8\sqrt{257}}{-2} when ± is minus. Subtract 8\sqrt{257} from 130.

$$x=\frac{130-8\sqrt{257}}{-2}$$

Divide 130-8\sqrt{257} by -2.

$$x=4\sqrt{257}-65$$

The equation is now solved.

$$x=-4\sqrt{257}-65 x=4\sqrt{257}-65$$

Substitute -4\sqrt{257}-65 for x in the equation -2\sqrt{6x^{2}-34+48}=-13-5x.

$$-2\sqrt{6\left(-4\sqrt{257}-65\right)^{2}-34+48}=-13-5\left(-4\sqrt{257}-65\right)$$

Simplify. The value x=-4\sqrt{257}-65 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-20\times 257^{\frac{1}{2}}-312=312+20\times 257^{\frac{1}{2}}$$

Substitute 4\sqrt{257}-65 for x in the equation -2\sqrt{6x^{2}-34+48}=-13-5x.

$$-2\sqrt{6\left(4\sqrt{257}-65\right)^{2}-34+48}=-13-5\left(4\sqrt{257}-65\right)$$

Simplify. The value x=4\sqrt{257}-65 satisfies the equation.

$$-20\times 257^{\frac{1}{2}}+312=312-20\times 257^{\frac{1}{2}}$$

Equation -2\sqrt{6x^{2}+14}=-5x-13 has a unique solution.

$$x=4\sqrt{257}-65$$