Solve 8sqrt{100-x^2}=6x+60

$$8 \sqrt { 100 - x ^ { 2 } } = 6 x + 60$$

Solve for x

$$x = \frac{14}{5} = 2\frac{4}{5} = 2.8$$

$$x=-10$$

Solution Steps

Square both sides of the equation.

$$\left(8\sqrt{100-x^{2}}\right)^{2}=\left(6x+60\right)^{2}$$

Expand \left(8\sqrt{100-x^{2}}\right)^{2}.

$$8^{2}\left(\sqrt{100-x^{2}}\right)^{2}=\left(6x+60\right)^{2}$$

Calculate 8 to the power of 2 and get 64.

$$64\left(\sqrt{100-x^{2}}\right)^{2}=\left(6x+60\right)^{2}$$

Calculate \sqrt{100-x^{2}} to the power of 2 and get 100-x^{2}.

$$64\left(100-x^{2}\right)=\left(6x+60\right)^{2}$$

Use the distributive property to multiply 64 by 100-x^{2}.

$$6400-64x^{2}=\left(6x+60\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6x+60\right)^{2}.

$$6400-64x^{2}=36x^{2}+720x+3600$$

Subtract 36x^{2} from both sides.

$$6400-64x^{2}-36x^{2}=720x+3600$$

Combine -64x^{2} and -36x^{2} to get -100x^{2}.

$$6400-100x^{2}=720x+3600$$

Subtract 720x from both sides.

$$6400-100x^{2}-720x=3600$$

Subtract 3600 from both sides.

$$6400-100x^{2}-720x-3600=0$$

Subtract 3600 from 6400 to get 2800.

$$2800-100x^{2}-720x=0$$

Divide both sides by 20.

$$140-5x^{2}-36x=0$$

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

$$-5x^{2}-36x+140=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -5x^{2}+ax+bx+140. To find a and b, set up a system to be solved.

$$a+b=-36 ab=-5\times 140=-700$$

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 -700.

$$1,-700 2,-350 4,-175 5,-140 7,-100 10,-70 14,-50 20,-35 25,-28$$

Calculate the sum for each pair.

$$1-700=-699 2-350=-348 4-175=-171 5-140=-135 7-100=-93 10-70=-60 14-50=-36 20-35=-15 25-28=-3$$

The solution is the pair that gives sum -36.

$$a=14 b=-50$$

Rewrite -5x^{2}-36x+140 as \left(-5x^{2}+14x\right)+\left(-50x+140\right).

$$\left(-5x^{2}+14x\right)+\left(-50x+140\right)$$

Factor out -x in the first and -10 in the second group.

$$-x\left(5x-14\right)-10\left(5x-14\right)$$

Factor out common term 5x-14 by using distributive property.

$$\left(5x-14\right)\left(-x-10\right)$$

To find equation solutions, solve 5x-14=0 and -x-10=0.

$$x=\frac{14}{5} x=-10$$

Substitute \frac{14}{5} for x in the equation 8\sqrt{100-x^{2}}=6x+60.

$$8\sqrt{100-\left(\frac{14}{5}\right)^{2}}=6\times \frac{14}{5}+60$$

Simplify. The value x=\frac{14}{5} satisfies the equation.

$$\frac{384}{5}=\frac{384}{5}$$

Substitute -10 for x in the equation 8\sqrt{100-x^{2}}=6x+60.

$$8\sqrt{100-\left(-10\right)^{2}}=6\left(-10\right)+60$$

Simplify. The value x=-10 satisfies the equation.

$$0=0$$

List all solutions of 8\sqrt{100-x^{2}}=6x+60.

$$x=\frac{14}{5} x=-10$$