Solve 0.766^2=x^2+1.1547^2-2x*1.1547cos(41.1)

$${ 0.766 }^{ 2 } = { x }^{ 2 } + { 1.1547 }^{ 2 } -2x \times 1.1547 \cos ( 41.1 )$$

Solve for x

$$x=\frac{\sqrt{105669189196887663414023655014284225969}+87013964909070128313}{100000000000000000000}\approx 0.972935169$$

$$x=\frac{87013964909070128313-\sqrt{105669189196887663414023655014284225969}}{100000000000000000000}\approx 0.767344129$$

Steps Using the Quadratic Formula

Evaluate trigonometric functions in the problem

$${(0.766)} ^ {2} = {(x)} ^ {2} + {(1.1547)} ^ {2} - 2 x \cdot 1.1547 \cdot 0.7535633923016379$$

Calculate 0.766 to the power of 2 and get 0.586756.

$$0.586756=x^{2}+1.1547^{2}-2x\times 1.1547\times 0.7535633923016379$$

Calculate 1.1547 to the power of 2 and get 1.33333209.

$$0.586756=x^{2}+1.33333209-2x\times 1.1547\times 0.7535633923016379$$

Multiply 2 and 1.1547 to get 2.3094.

$$0.586756=x^{2}+1.33333209-2.3094x\times 0.7535633923016379$$

Multiply 2.3094 and 0.7535633923016379 to get 1.74027929818140256626.

$$0.586756=x^{2}+1.33333209-1.74027929818140256626x$$

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

$$x^{2}+1.33333209-1.74027929818140256626x=0.586756$$

Subtract 0.586756 from both sides.

$$x^{2}+1.33333209-1.74027929818140256626x-0.586756=0$$

Subtract 0.586756 from 1.33333209 to get 0.74657609.

$$x^{2}+0.74657609-1.74027929818140256626x=0$$

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.

$$x^{2}-1.74027929818140256626x+0.74657609=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1.74027929818140256626 for b, and 0.74657609 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1.74027929818140256626\right)±\sqrt{\left(-1.74027929818140256626\right)^{2}-4\times 0.74657609}}{2}$$

Square -1.74027929818140256626 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-1.74027929818140256626\right)±\sqrt{3.0285720356787550653656094620057136903876-4\times 0.74657609}}{2}$$

Multiply -4 times 0.74657609.

$$x=\frac{-\left(-1.74027929818140256626\right)±\sqrt{3.0285720356787550653656094620057136903876-2.98630436}}{2}$$

Add 3.0285720356787550653656094620057136903876 to -2.98630436 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-1.74027929818140256626\right)±\sqrt{0.0422676756787550653656094620057136903876}}{2}$$

Take the square root of 0.0422676756787550653656094620057136903876.

$$x=\frac{-\left(-1.74027929818140256626\right)±\frac{\sqrt{105669189196887663414023655014284225969}}{50000000000000000000}}{2}$$

The opposite of -1.74027929818140256626 is 1.74027929818140256626.

$$x=\frac{1.74027929818140256626±\frac{\sqrt{105669189196887663414023655014284225969}}{50000000000000000000}}{2}$$

Now solve the equation x=\frac{1.74027929818140256626±\frac{\sqrt{105669189196887663414023655014284225969}}{50000000000000000000}}{2} when ± is plus. Add 1.74027929818140256626 to \frac{\sqrt{105669189196887663414023655014284225969}}{50000000000000000000}.

$$x=\frac{\sqrt{105669189196887663414023655014284225969}+87013964909070128313}{2\times 50000000000000000000}$$

Divide \frac{87013964909070128313+\sqrt{105669189196887663414023655014284225969}}{50000000000000000000} by 2.

$$x=\frac{\sqrt{105669189196887663414023655014284225969}+87013964909070128313}{100000000000000000000}$$

Now solve the equation x=\frac{1.74027929818140256626±\frac{\sqrt{105669189196887663414023655014284225969}}{50000000000000000000}}{2} when ± is minus. Subtract \frac{\sqrt{105669189196887663414023655014284225969}}{50000000000000000000} from 1.74027929818140256626.

$$x=\frac{87013964909070128313-\sqrt{105669189196887663414023655014284225969}}{2\times 50000000000000000000}$$

Divide \frac{87013964909070128313-\sqrt{105669189196887663414023655014284225969}}{50000000000000000000} by 2.

$$x=\frac{87013964909070128313-\sqrt{105669189196887663414023655014284225969}}{100000000000000000000}$$

The equation is now solved.

$$x=\frac{\sqrt{105669189196887663414023655014284225969}+87013964909070128313}{100000000000000000000} x=\frac{87013964909070128313-\sqrt{105669189196887663414023655014284225969}}{100000000000000000000}$$

Steps for Completing the Square

Evaluate trigonometric functions in the problem

$${(0.766)} ^ {2} = {(x)} ^ {2} + {(1.1547)} ^ {2} - 2 x \cdot 1.1547 \cdot 0.7535633923016379$$

Calculate 0.766 to the power of 2 and get 0.586756.

$$0.586756=x^{2}+1.1547^{2}-2x\times 1.1547\times 0.7535633923016379$$

Calculate 1.1547 to the power of 2 and get 1.33333209.

$$0.586756=x^{2}+1.33333209-2x\times 1.1547\times 0.7535633923016379$$

Multiply 2 and 1.1547 to get 2.3094.

$$0.586756=x^{2}+1.33333209-2.3094x\times 0.7535633923016379$$

Multiply 2.3094 and 0.7535633923016379 to get 1.74027929818140256626.

$$0.586756=x^{2}+1.33333209-1.74027929818140256626x$$

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

$$x^{2}+1.33333209-1.74027929818140256626x=0.586756$$

Subtract 1.33333209 from both sides.

$$x^{2}-1.74027929818140256626x=0.586756-1.33333209$$

Subtract 1.33333209 from 0.586756 to get -0.74657609.

$$x^{2}-1.74027929818140256626x=-0.74657609$$

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.

$$x^{2}-1.74027929818140256626x=-\frac{74657609}{100000000}$$

Divide -1.74027929818140256626, the coefficient of the x term, by 2 to get -0.87013964909070128313. Then add the square of -0.87013964909070128313 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1.74027929818140256626x+\left(-0.87013964909070128313\right)^{2}=-\frac{74657609}{100000000}+\left(-0.87013964909070128313\right)^{2}$$

Square -0.87013964909070128313 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-1.74027929818140256626x+0.7571430089196887663414023655014284225969=-\frac{74657609}{100000000}+0.7571430089196887663414023655014284225969$$

Add -\frac{74657609}{100000000} to 0.7571430089196887663414023655014284225969 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-1.74027929818140256626x+0.7571430089196887663414023655014284225969=\frac{105669189196887663414023655014284225969}{10000000000000000000000000000000000000000}$$

Factor x^{2}-1.74027929818140256626x+0.7571430089196887663414023655014284225969. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.87013964909070128313\right)^{2}=\frac{105669189196887663414023655014284225969}{10000000000000000000000000000000000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.87013964909070128313\right)^{2}}=\sqrt{\frac{105669189196887663414023655014284225969}{10000000000000000000000000000000000000000}}$$

Simplify.

$$x-0.87013964909070128313=\frac{\sqrt{105669189196887663414023655014284225969}}{100000000000000000000} x-0.87013964909070128313=-\frac{\sqrt{105669189196887663414023655014284225969}}{100000000000000000000}$$

Add 0.87013964909070128313 to both sides of the equation.

$$x=\frac{\sqrt{105669189196887663414023655014284225969}+87013964909070128313}{100000000000000000000} x=\frac{87013964909070128313-\sqrt{105669189196887663414023655014284225969}}{100000000000000000000}$$