Solve (x+5)^2/x+2=32cos22.5

$$\frac { ( x + 5 ) ^ { 2 } } { x + 2 } = 32 \cos 22.5$$

Solve for x

$$x = \frac{27 \sqrt{69560845802680541287937296241} + 6113795325112867}{625000000000000} = 21\frac{109880677090760}{625000000000000} \approx 21.175809083$$

$$x=\frac{6113795325112867-27\sqrt{69560845802680541287937296241}}{625000000000000}\approx -1.611664043$$

Steps Using the Quadratic Formula

Evaluate trigonometric functions in the problem

$$\frac{{(x + 5)} ^ {2}}{x + 2} = 32 \cdot 0.9238795325112867$$

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$\left(x+5\right)^{2}=32\times 0.9238795325112867\left(x+2\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+5\right)^{2}.

$$x^{2}+10x+25=32\times 0.9238795325112867\left(x+2\right)$$

Multiply 32 and 0.9238795325112867 to get 29.5641450403611744.

$$x^{2}+10x+25=29.5641450403611744\left(x+2\right)$$

Use the distributive property to multiply 29.5641450403611744 by x+2.

$$x^{2}+10x+25=29.5641450403611744x+59.1282900807223488$$

Subtract 29.5641450403611744x from both sides.

$$x^{2}+10x+25-29.5641450403611744x=59.1282900807223488$$

Combine 10x and -29.5641450403611744x to get -19.5641450403611744x.

$$x^{2}-19.5641450403611744x+25=59.1282900807223488$$

Subtract 59.1282900807223488 from both sides.

$$x^{2}-19.5641450403611744x+25-59.1282900807223488=0$$

Subtract 59.1282900807223488 from 25 to get -34.1282900807223488.

$$x^{2}-19.5641450403611744x-34.1282900807223488=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -19.5641450403611744 for b, and -34.1282900807223488 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-19.5641450403611744\right)±\sqrt{\left(-19.5641450403611744\right)^{2}-4\left(-34.1282900807223488\right)}}{2}$$

Square -19.5641450403611744 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-19.5641450403611744\right)±\sqrt{382.75577116028873829280039894721536-4\left(-34.1282900807223488\right)}}{2}$$

Multiply -4 times -34.1282900807223488.

$$x=\frac{-\left(-19.5641450403611744\right)±\sqrt{382.75577116028873829280039894721536+136.5131603228893952}}{2}$$

Add 382.75577116028873829280039894721536 to 136.5131603228893952 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-19.5641450403611744\right)±\sqrt{519.26893148317813349280039894721536}}{2}$$

Take the square root of 519.26893148317813349280039894721536.

$$x=\frac{-\left(-19.5641450403611744\right)±\frac{27\sqrt{69560845802680541287937296241}}{312500000000000}}{2}$$

The opposite of -19.5641450403611744 is 19.5641450403611744.

$$x=\frac{19.5641450403611744±\frac{27\sqrt{69560845802680541287937296241}}{312500000000000}}{2}$$

Now solve the equation x=\frac{19.5641450403611744±\frac{27\sqrt{69560845802680541287937296241}}{312500000000000}}{2} when ± is plus. Add 19.5641450403611744 to \frac{27\sqrt{69560845802680541287937296241}}{312500000000000}.

$$x=\frac{27\sqrt{69560845802680541287937296241}+6113795325112867}{2\times 312500000000000}$$

Divide \frac{6113795325112867+27\sqrt{69560845802680541287937296241}}{312500000000000} by 2.

$$x=\frac{27\sqrt{69560845802680541287937296241}+6113795325112867}{625000000000000}$$

Now solve the equation x=\frac{19.5641450403611744±\frac{27\sqrt{69560845802680541287937296241}}{312500000000000}}{2} when ± is minus. Subtract \frac{27\sqrt{69560845802680541287937296241}}{312500000000000} from 19.5641450403611744.

$$x=\frac{6113795325112867-27\sqrt{69560845802680541287937296241}}{2\times 312500000000000}$$

Divide \frac{6113795325112867-27\sqrt{69560845802680541287937296241}}{312500000000000} by 2.

$$x=\frac{6113795325112867-27\sqrt{69560845802680541287937296241}}{625000000000000}$$

The equation is now solved.

$$x=\frac{27\sqrt{69560845802680541287937296241}+6113795325112867}{625000000000000} x=\frac{6113795325112867-27\sqrt{69560845802680541287937296241}}{625000000000000}$$

Steps for Completing the Square

Evaluate trigonometric functions in the problem

$$\frac{{(x + 5)} ^ {2}}{x + 2} = 32 \cdot 0.9238795325112867$$

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$\left(x+5\right)^{2}=32\times 0.9238795325112867\left(x+2\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+5\right)^{2}.

$$x^{2}+10x+25=32\times 0.9238795325112867\left(x+2\right)$$

Multiply 32 and 0.9238795325112867 to get 29.5641450403611744.

$$x^{2}+10x+25=29.5641450403611744\left(x+2\right)$$

Use the distributive property to multiply 29.5641450403611744 by x+2.

$$x^{2}+10x+25=29.5641450403611744x+59.1282900807223488$$

Subtract 29.5641450403611744x from both sides.

$$x^{2}+10x+25-29.5641450403611744x=59.1282900807223488$$

Combine 10x and -29.5641450403611744x to get -19.5641450403611744x.

$$x^{2}-19.5641450403611744x+25=59.1282900807223488$$

Subtract 25 from both sides.

$$x^{2}-19.5641450403611744x=59.1282900807223488-25$$

Subtract 25 from 59.1282900807223488 to get 34.1282900807223488.

$$x^{2}-19.5641450403611744x=34.1282900807223488$$

Divide -19.5641450403611744, the coefficient of the x term, by 2 to get -9.7820725201805872. Then add the square of -9.7820725201805872 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-19.5641450403611744x+\left(-9.7820725201805872\right)^{2}=34.1282900807223488+\left(-9.7820725201805872\right)^{2}$$

Square -9.7820725201805872 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-19.5641450403611744x+95.68894279007218457320009973680384=34.1282900807223488+95.68894279007218457320009973680384$$

Add 34.1282900807223488 to 95.68894279007218457320009973680384 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-19.5641450403611744x+95.68894279007218457320009973680384=129.81723287079453337320009973680384$$

Factor x^{2}-19.5641450403611744x+95.68894279007218457320009973680384. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-9.7820725201805872\right)^{2}=129.81723287079453337320009973680384$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9.7820725201805872\right)^{2}}=\sqrt{129.81723287079453337320009973680384}$$

Simplify.

$$x-9.7820725201805872=\frac{27\sqrt{69560845802680541287937296241}}{625000000000000} x-9.7820725201805872=-\frac{27\sqrt{69560845802680541287937296241}}{625000000000000}$$

Add 9.7820725201805872 to both sides of the equation.

$$x=\frac{27\sqrt{69560845802680541287937296241}+6113795325112867}{625000000000000} x=\frac{6113795325112867-27\sqrt{69560845802680541287937296241}}{625000000000000}$$