Solve 2c-17=sqrt{-121+13c}

$$2 c - 17 = \sqrt { - 121 + 13 c }$$

Solve for c

$$c = \frac{41}{4} = 10\frac{1}{4} = 10.25$$

$$c=10$$

Solution Steps

Square both sides of the equation.

$$\left(2c-17\right)^{2}=\left(\sqrt{-121+13c}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2c-17\right)^{2}.

$$4c^{2}-68c+289=\left(\sqrt{-121+13c}\right)^{2}$$

Calculate \sqrt{-121+13c} to the power of 2 and get -121+13c.

$$4c^{2}-68c+289=-121+13c$$

Subtract -121 from both sides.

$$4c^{2}-68c+289-\left(-121\right)=13c$$

The opposite of -121 is 121.

$$4c^{2}-68c+289+121=13c$$

Subtract 13c from both sides.

$$4c^{2}-68c+289+121-13c=0$$

Add 289 and 121 to get 410.

$$4c^{2}-68c+410-13c=0$$

Combine -68c and -13c to get -81c.

$$4c^{2}-81c+410=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -81 for b, and 410 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{-\left(-81\right)±\sqrt{\left(-81\right)^{2}-4\times 4\times 410}}{2\times 4}$$

Square -81.

$$c=\frac{-\left(-81\right)±\sqrt{6561-4\times 4\times 410}}{2\times 4}$$

Multiply -4 times 4.

$$c=\frac{-\left(-81\right)±\sqrt{6561-16\times 410}}{2\times 4}$$

Multiply -16 times 410.

$$c=\frac{-\left(-81\right)±\sqrt{6561-6560}}{2\times 4}$$

Add 6561 to -6560.

$$c=\frac{-\left(-81\right)±\sqrt{1}}{2\times 4}$$

Take the square root of 1.

$$c=\frac{-\left(-81\right)±1}{2\times 4}$$

The opposite of -81 is 81.

$$c=\frac{81±1}{2\times 4}$$

Multiply 2 times 4.

$$c=\frac{81±1}{8}$$

Now solve the equation c=\frac{81±1}{8} when ± is plus. Add 81 to 1.

$$c=\frac{82}{8}$$

Reduce the fraction \frac{82}{8} to lowest terms by extracting and canceling out 2.

$$c=\frac{41}{4}$$

Now solve the equation c=\frac{81±1}{8} when ± is minus. Subtract 1 from 81.

$$c=\frac{80}{8}$$

Divide 80 by 8.

$$c=10$$

The equation is now solved.

$$c=\frac{41}{4} c=10$$

Substitute \frac{41}{4} for c in the equation 2c-17=\sqrt{-121+13c}.

$$2\times \frac{41}{4}-17=\sqrt{-121+13\times \frac{41}{4}}$$

Simplify. The value c=\frac{41}{4} satisfies the equation.

$$\frac{7}{2}=\frac{7}{2}$$

Substitute 10 for c in the equation 2c-17=\sqrt{-121+13c}.

$$2\times 10-17=\sqrt{-121+13\times 10}$$

Simplify. The value c=10 satisfies the equation.

$$3=3$$

List all solutions of 2c-17=\sqrt{13c-121}.

$$c=\frac{41}{4} c=10$$