Solve sqrt{17x-2x^2-21}=120/12-14x/12

$$\sqrt { 17 x - 2 x ^ { 2 } - 21 } = \frac { 120 } { 12 } - \frac { 14 x } { 12 }$$

Solve for x

$$x=6$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{17x-2x^{2}-21}\right)^{2}=\left(\frac{120}{12}-\frac{14x}{12}\right)^{2}$$

Calculate \sqrt{17x-2x^{2}-21} to the power of 2 and get 17x-2x^{2}-21.

$$17x-2x^{2}-21=\left(\frac{120}{12}-\frac{14x}{12}\right)^{2}$$

Divide 120 by 12 to get 10.

$$17x-2x^{2}-21=\left(10-\frac{14x}{12}\right)^{2}$$

Divide 14x by 12 to get \frac{7}{6}x.

$$17x-2x^{2}-21=\left(10-\frac{7}{6}x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(10-\frac{7}{6}x\right)^{2}.

$$17x-2x^{2}-21=100+20\left(-\frac{7}{6}x\right)+\left(-\frac{7}{6}x\right)^{2}$$

Calculate -\frac{7}{6}x to the power of 2 and get \left(\frac{7}{6}x\right)^{2}.

$$17x-2x^{2}-21=100+20\left(-\frac{7}{6}x\right)+\left(\frac{7}{6}x\right)^{2}$$

Expand \left(\frac{7}{6}x\right)^{2}.

$$17x-2x^{2}-21=100+20\left(-\frac{7}{6}x\right)+\left(\frac{7}{6}\right)^{2}x^{2}$$

Calculate \frac{7}{6} to the power of 2 and get \frac{49}{36}.

$$17x-2x^{2}-21=100+20\left(-\frac{7}{6}x\right)+\frac{49}{36}x^{2}$$

Subtract 100 from both sides.

$$17x-2x^{2}-21-100=20\left(-\frac{7}{6}x\right)+\frac{49}{36}x^{2}$$

Subtract 100 from -21 to get -121.

$$17x-2x^{2}-121=20\left(-\frac{7}{6}x\right)+\frac{49}{36}x^{2}$$

Subtract 20\left(-\frac{7}{6}x\right) from both sides.

$$17x-2x^{2}-121-20\left(-\frac{7}{6}x\right)=\frac{49}{36}x^{2}$$

Subtract \frac{49}{36}x^{2} from both sides.

$$17x-2x^{2}-121-20\left(-\frac{7}{6}x\right)-\frac{49}{36}x^{2}=0$$

Multiply -1 and 20 to get -20.

$$17x-2x^{2}-121-20\left(-1\right)\times \frac{7}{6}x-\frac{49}{36}x^{2}=0$$

Multiply -20 and -1 to get 20.

$$17x-2x^{2}-121+20\times \frac{7}{6}x-\frac{49}{36}x^{2}=0$$

Multiply 20 and \frac{7}{6} to get \frac{70}{3}.

$$17x-2x^{2}-121+\frac{70}{3}x-\frac{49}{36}x^{2}=0$$

Combine 17x and \frac{70}{3}x to get \frac{121}{3}x.

$$\frac{121}{3}x-2x^{2}-121-\frac{49}{36}x^{2}=0$$

Combine -2x^{2} and -\frac{49}{36}x^{2} to get -\frac{121}{36}x^{2}.

$$\frac{121}{3}x-\frac{121}{36}x^{2}-121=0$$

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.

$$-\frac{121}{36}x^{2}+\frac{121}{3}x-121=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{121}{36} for a, \frac{121}{3} for b, and -121 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{121}{3}±\sqrt{\left(\frac{121}{3}\right)^{2}-4\left(-\frac{121}{36}\right)\left(-121\right)}}{2\left(-\frac{121}{36}\right)}$$

Square \frac{121}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{121}{3}±\sqrt{\frac{14641}{9}-4\left(-\frac{121}{36}\right)\left(-121\right)}}{2\left(-\frac{121}{36}\right)}$$

Multiply -4 times -\frac{121}{36}.

$$x=\frac{-\frac{121}{3}±\sqrt{\frac{14641}{9}+\frac{121}{9}\left(-121\right)}}{2\left(-\frac{121}{36}\right)}$$

Multiply \frac{121}{9} times -121.

$$x=\frac{-\frac{121}{3}±\sqrt{\frac{14641-14641}{9}}}{2\left(-\frac{121}{36}\right)}$$

Add \frac{14641}{9} to -\frac{14641}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{121}{3}±\sqrt{0}}{2\left(-\frac{121}{36}\right)}$$

Take the square root of 0.

$$x=-\frac{\frac{121}{3}}{2\left(-\frac{121}{36}\right)}$$

Multiply 2 times -\frac{121}{36}.

$$x=-\frac{\frac{121}{3}}{-\frac{121}{18}}$$

Divide -\frac{121}{3} by -\frac{121}{18} by multiplying -\frac{121}{3} by the reciprocal of -\frac{121}{18}.

$$x=6$$

Substitute 6 for x in the equation \sqrt{17x-2x^{2}-21}=\frac{120}{12}-\frac{14x}{12}.

$$\sqrt{17\times 6-2\times 6^{2}-21}=\frac{120}{12}-\frac{14\times 6}{12}$$

Simplify. The value x=6 satisfies the equation.

$$3=3$$

Equation \sqrt{-2x^{2}+17x-21}=-\frac{14x}{12}+10 has a unique solution.

$$x=6$$