Solve 1/7x-frac{4sqrt{3}}{7}sqrt{1-x^2}=-11/14

$$\frac { 1 } { 7 } x - \frac { 4 \sqrt { 3 } } { 7 } \sqrt { 1 - x ^ { 2 } } = - \frac { 11 } { 14 }$$

Solve for x

$$x=\frac{1}{2}=0.5$$

$$x=-\frac{71}{98}\approx -0.724489796$$

Solution Steps

Subtract \frac{1}{7}x from both sides of the equation.

$$-\frac{4\sqrt{3}}{7}\sqrt{1-x^{2}}=-\frac{11}{14}-\frac{1}{7}x$$

Multiply both sides of the equation by 14, the least common multiple of 7,14.

$$-14\times \frac{4\sqrt{3}}{7}\sqrt{1-x^{2}}=-11-2x$$

Cancel out 7, the greatest common factor in 14 and 7.

$$-2\times 4\sqrt{3}\sqrt{1-x^{2}}=-11-2x$$

Multiply -2 and 4 to get -8.

$$-8\sqrt{3}\sqrt{1-x^{2}}=-11-2x$$

Square both sides of the equation.

$$\left(-8\sqrt{3}\sqrt{1-x^{2}}\right)^{2}=\left(-11-2x\right)^{2}$$

Expand \left(-8\sqrt{3}\sqrt{1-x^{2}}\right)^{2}.

$$\left(-8\right)^{2}\left(\sqrt{3}\right)^{2}\left(\sqrt{1-x^{2}}\right)^{2}=\left(-11-2x\right)^{2}$$

Calculate -8 to the power of 2 and get 64.

$$64\left(\sqrt{3}\right)^{2}\left(\sqrt{1-x^{2}}\right)^{2}=\left(-11-2x\right)^{2}$$

The square of \sqrt{3} is 3.

$$64\times 3\left(\sqrt{1-x^{2}}\right)^{2}=\left(-11-2x\right)^{2}$$

Multiply 64 and 3 to get 192.

$$192\left(\sqrt{1-x^{2}}\right)^{2}=\left(-11-2x\right)^{2}$$

Calculate \sqrt{1-x^{2}} to the power of 2 and get 1-x^{2}.

$$192\left(1-x^{2}\right)=\left(-11-2x\right)^{2}$$

Use the distributive property to multiply 192 by 1-x^{2}.

$$192-192x^{2}=\left(-11-2x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-11-2x\right)^{2}.

$$192-192x^{2}=121+44x+4x^{2}$$

Subtract 121 from both sides.

$$192-192x^{2}-121=44x+4x^{2}$$

Subtract 121 from 192 to get 71.

$$71-192x^{2}=44x+4x^{2}$$

Subtract 44x from both sides.

$$71-192x^{2}-44x=4x^{2}$$

Subtract 4x^{2} from both sides.

$$71-192x^{2}-44x-4x^{2}=0$$

Combine -192x^{2} and -4x^{2} to get -196x^{2}.

$$71-196x^{2}-44x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-196x^{2}-44x+71=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -196x^{2}+ax+bx+71. To find a and b, set up a system to be solved.

$$a+b=-44 ab=-196\times 71=-13916$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -13916.

$$1,-13916 2,-6958 4,-3479 7,-1988 14,-994 28,-497 49,-284 71,-196 98,-142$$

Calculate the sum for each pair.

$$1-13916=-13915 2-6958=-6956 4-3479=-3475 7-1988=-1981 14-994=-980 28-497=-469 49-284=-235 71-196=-125 98-142=-44$$

The solution is the pair that gives sum -44.

$$a=98 b=-142$$

Rewrite -196x^{2}-44x+71 as \left(-196x^{2}+98x\right)+\left(-142x+71\right).

$$\left(-196x^{2}+98x\right)+\left(-142x+71\right)$$

Factor out -98x in the first and -71 in the second group.

$$-98x\left(2x-1\right)-71\left(2x-1\right)$$

Factor out common term 2x-1 by using distributive property.

$$\left(2x-1\right)\left(-98x-71\right)$$

To find equation solutions, solve 2x-1=0 and -98x-71=0.

$$x=\frac{1}{2} x=-\frac{71}{98}$$

Substitute \frac{1}{2} for x in the equation \frac{1}{7}x-\frac{4\sqrt{3}}{7}\sqrt{1-x^{2}}=-\frac{11}{14}.

$$\frac{1}{7}\times \frac{1}{2}-\frac{4\sqrt{3}}{7}\sqrt{1-\left(\frac{1}{2}\right)^{2}}=-\frac{11}{14}$$

Simplify. The value x=\frac{1}{2} satisfies the equation.

$$-\frac{11}{14}=-\frac{11}{14}$$

Substitute -\frac{71}{98} for x in the equation \frac{1}{7}x-\frac{4\sqrt{3}}{7}\sqrt{1-x^{2}}=-\frac{11}{14}.

$$\frac{1}{7}\left(-\frac{71}{98}\right)-\frac{4\sqrt{3}}{7}\sqrt{1-\left(-\frac{71}{98}\right)^{2}}=-\frac{11}{14}$$

Simplify. The value x=-\frac{71}{98} satisfies the equation.

$$-\frac{11}{14}=-\frac{11}{14}$$

List all solutions of -8\sqrt{3}\sqrt{1-x^{2}}=-2x-11.

$$x=\frac{1}{2} x=-\frac{71}{98}$$