Solve 12.12^2=10.8^2+x^2-2*10.8*x*cos122^circ

$$12.12 ^ { 2 } = 10.8 ^ { 2 } + x ^ { 2 } - 2 \cdot 10.8 \cdot x \cdot \cos 122 ^ { \circ }$$

Solve for x

$$x = \frac{3 \sqrt{17092175216813418503407198897729} - 8942387583935331}{1562500000000000} = 2\frac{335417620989915}{1562500000000000} \approx 2.214667277$$

$$x=\frac{-3\sqrt{17092175216813418503407198897729}-8942387583935331}{1562500000000000}\approx -13.660923385$$

Steps Using the Quadratic Formula

Evaluate trigonometric functions in the problem

$$12.12 ^ {2} = 10.8 ^ {2} + x ^ {2} - 2 \cdot 10.8 \cdot x \cdot -0.5299192642332048$$

Calculate 12.12 to the power of 2 and get 146.8944.

$$146.8944=10.8^{2}+x^{2}-2\times 10.8x\left(-0.5299192642332048\right)$$

Calculate 10.8 to the power of 2 and get 116.64.

$$146.8944=116.64+x^{2}-2\times 10.8x\left(-0.5299192642332048\right)$$

Multiply 2 and 10.8 to get 21.6.

$$146.8944=116.64+x^{2}-21.6x\left(-0.5299192642332048\right)$$

Multiply 21.6 and -0.5299192642332048 to get -11.44625610743722368.

$$146.8944=116.64+x^{2}-\left(-11.44625610743722368x\right)$$

The opposite of -11.44625610743722368x is 11.44625610743722368x.

$$146.8944=116.64+x^{2}+11.44625610743722368x$$

Swap sides so that all variable terms are on the left hand side.

$$116.64+x^{2}+11.44625610743722368x=146.8944$$

Subtract 146.8944 from both sides.

$$116.64+x^{2}+11.44625610743722368x-146.8944=0$$

Subtract 146.8944 from 116.64 to get -30.2544.

$$-30.2544+x^{2}+11.44625610743722368x=0$$

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.

$$x^{2}+11.44625610743722368x-30.2544=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 11.44625610743722368 for b, and -30.2544 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11.44625610743722368±\sqrt{11.44625610743722368^{2}-4\left(-30.2544\right)}}{2}$$

Square 11.44625610743722368 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-11.44625610743722368±\sqrt{131.0167788770439438838411920663527424-4\left(-30.2544\right)}}{2}$$

Multiply -4 times -30.2544.

$$x=\frac{-11.44625610743722368±\sqrt{131.0167788770439438838411920663527424+121.0176}}{2}$$

Add 131.0167788770439438838411920663527424 to 121.0176 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-11.44625610743722368±\sqrt{252.0343788770439438838411920663527424}}{2}$$

Take the square root of 252.0343788770439438838411920663527424.

$$x=\frac{-11.44625610743722368±\frac{3\sqrt{17092175216813418503407198897729}}{781250000000000}}{2}$$

Now solve the equation x=\frac{-11.44625610743722368±\frac{3\sqrt{17092175216813418503407198897729}}{781250000000000}}{2} when ± is plus. Add -11.44625610743722368 to \frac{3\sqrt{17092175216813418503407198897729}}{781250000000000}.

$$x=\frac{3\sqrt{17092175216813418503407198897729}-8942387583935331}{2\times 781250000000000}$$

Divide \frac{-8942387583935331+3\sqrt{17092175216813418503407198897729}}{781250000000000} by 2.

$$x=\frac{3\sqrt{17092175216813418503407198897729}-8942387583935331}{1562500000000000}$$

Now solve the equation x=\frac{-11.44625610743722368±\frac{3\sqrt{17092175216813418503407198897729}}{781250000000000}}{2} when ± is minus. Subtract \frac{3\sqrt{17092175216813418503407198897729}}{781250000000000} from -11.44625610743722368.

$$x=\frac{-3\sqrt{17092175216813418503407198897729}-8942387583935331}{2\times 781250000000000}$$

Divide \frac{-8942387583935331-3\sqrt{17092175216813418503407198897729}}{781250000000000} by 2.

$$x=\frac{-3\sqrt{17092175216813418503407198897729}-8942387583935331}{1562500000000000}$$

The equation is now solved.

$$x=\frac{3\sqrt{17092175216813418503407198897729}-8942387583935331}{1562500000000000} x=\frac{-3\sqrt{17092175216813418503407198897729}-8942387583935331}{1562500000000000}$$

Steps for Completing the Square

Evaluate trigonometric functions in the problem

$$12.12 ^ {2} = 10.8 ^ {2} + x ^ {2} - 2 \cdot 10.8 \cdot x \cdot -0.5299192642332048$$

Calculate 12.12 to the power of 2 and get 146.8944.

$$146.8944=10.8^{2}+x^{2}-2\times 10.8x\left(-0.5299192642332048\right)$$

Calculate 10.8 to the power of 2 and get 116.64.

$$146.8944=116.64+x^{2}-2\times 10.8x\left(-0.5299192642332048\right)$$

Multiply 2 and 10.8 to get 21.6.

$$146.8944=116.64+x^{2}-21.6x\left(-0.5299192642332048\right)$$

Multiply 21.6 and -0.5299192642332048 to get -11.44625610743722368.

$$146.8944=116.64+x^{2}-\left(-11.44625610743722368x\right)$$

The opposite of -11.44625610743722368x is 11.44625610743722368x.

$$146.8944=116.64+x^{2}+11.44625610743722368x$$

Swap sides so that all variable terms are on the left hand side.

$$116.64+x^{2}+11.44625610743722368x=146.8944$$

Subtract 116.64 from both sides.

$$x^{2}+11.44625610743722368x=146.8944-116.64$$

Subtract 116.64 from 146.8944 to get 30.2544.

$$x^{2}+11.44625610743722368x=30.2544$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+11.44625610743722368x=\frac{18909}{625}$$

Divide 11.44625610743722368, the coefficient of the x term, by 2 to get 5.72312805371861184. Then add the square of 5.72312805371861184 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+11.44625610743722368x+5.72312805371861184^{2}=\frac{18909}{625}+5.72312805371861184^{2}$$

Square 5.72312805371861184 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+11.44625610743722368x+32.7541947192609859709602980165881856=\frac{18909}{625}+32.7541947192609859709602980165881856$$

Add \frac{18909}{625} to 32.7541947192609859709602980165881856 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+11.44625610743722368x+32.7541947192609859709602980165881856=\frac{153829576951320766530664790079561}{2441406250000000000000000000000}$$

Factor x^{2}+11.44625610743722368x+32.7541947192609859709602980165881856. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5.72312805371861184\right)^{2}=\frac{153829576951320766530664790079561}{2441406250000000000000000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5.72312805371861184\right)^{2}}=\sqrt{\frac{153829576951320766530664790079561}{2441406250000000000000000000000}}$$

Simplify.

$$x+5.72312805371861184=\frac{3\sqrt{17092175216813418503407198897729}}{1562500000000000} x+5.72312805371861184=-\frac{3\sqrt{17092175216813418503407198897729}}{1562500000000000}$$

Subtract 5.72312805371861184 from both sides of the equation.

$$x=\frac{3\sqrt{17092175216813418503407198897729}-8942387583935331}{1562500000000000} x=\frac{-3\sqrt{17092175216813418503407198897729}-8942387583935331}{1562500000000000}$$