Solve 60sqrt{L}-L-6=0

$$60 \sqrt { L } - L - 6 = 0$$

Solve for L

$$L=1794-60\sqrt{894}\approx 0.010033473$$

$$L=60\sqrt{894}+1794\approx 3587.989966527$$

Solution Steps

Subtract -L-6 from both sides of the equation.

$$60\sqrt{L}=-\left(-L-6\right)$$

To find the opposite of -L-6, find the opposite of each term.

$$60\sqrt{L}=-\left(-L\right)-\left(-6\right)$$

The opposite of -L is L.

$$60\sqrt{L}=L-\left(-6\right)$$

The opposite of -6 is 6.

$$60\sqrt{L}=L+6$$

Square both sides of the equation.

$$\left(60\sqrt{L}\right)^{2}=\left(L+6\right)^{2}$$

Expand \left(60\sqrt{L}\right)^{2}.

$$60^{2}\left(\sqrt{L}\right)^{2}=\left(L+6\right)^{2}$$

Calculate 60 to the power of 2 and get 3600.

$$3600\left(\sqrt{L}\right)^{2}=\left(L+6\right)^{2}$$

Calculate \sqrt{L} to the power of 2 and get L.

$$3600L=\left(L+6\right)^{2}$$

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

$$3600L=L^{2}+12L+36$$

Subtract L^{2} from both sides.

$$3600L-L^{2}=12L+36$$

Subtract 12L from both sides.

$$3600L-L^{2}-12L=36$$

Combine 3600L and -12L to get 3588L.

$$3588L-L^{2}=36$$

Subtract 36 from both sides.

$$3588L-L^{2}-36=0$$

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.

$$-L^{2}+3588L-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 3588 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$L=\frac{-3588±\sqrt{3588^{2}-4\left(-1\right)\left(-36\right)}}{2\left(-1\right)}$$

Square 3588.

$$L=\frac{-3588±\sqrt{12873744-4\left(-1\right)\left(-36\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$L=\frac{-3588±\sqrt{12873744+4\left(-36\right)}}{2\left(-1\right)}$$

Multiply 4 times -36.

$$L=\frac{-3588±\sqrt{12873744-144}}{2\left(-1\right)}$$

Add 12873744 to -144.

$$L=\frac{-3588±\sqrt{12873600}}{2\left(-1\right)}$$

Take the square root of 12873600.

$$L=\frac{-3588±120\sqrt{894}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$L=\frac{-3588±120\sqrt{894}}{-2}$$

Now solve the equation L=\frac{-3588±120\sqrt{894}}{-2} when ± is plus. Add -3588 to 120\sqrt{894}.

$$L=\frac{120\sqrt{894}-3588}{-2}$$

Divide -3588+120\sqrt{894} by -2.

$$L=1794-60\sqrt{894}$$

Now solve the equation L=\frac{-3588±120\sqrt{894}}{-2} when ± is minus. Subtract 120\sqrt{894} from -3588.

$$L=\frac{-120\sqrt{894}-3588}{-2}$$

Divide -3588-120\sqrt{894} by -2.

$$L=60\sqrt{894}+1794$$

The equation is now solved.

$$L=1794-60\sqrt{894} L=60\sqrt{894}+1794$$

Substitute 1794-60\sqrt{894} for L in the equation 60\sqrt{L}-L-6=0.

$$60\sqrt{1794-60\sqrt{894}}-\left(1794-60\sqrt{894}\right)-6=0$$

Simplify. The value L=1794-60\sqrt{894} satisfies the equation.

$$0=0$$

Substitute 60\sqrt{894}+1794 for L in the equation 60\sqrt{L}-L-6=0.

$$60\sqrt{60\sqrt{894}+1794}-\left(60\sqrt{894}+1794\right)-6=0$$

Simplify. The value L=60\sqrt{894}+1794 satisfies the equation.

$$0=0$$

List all solutions of 60\sqrt{L}=L+6.

$$L=1794-60\sqrt{894} L=60\sqrt{894}+1794$$