Solve 0.766^2=x^2+1.1547^2-2*1.1547*x*cos(38)

$${ 0.766 }^{ 2 } = { x }^{ 2 } + { 1.1547 }^{ 2 } -2 \cdot 1.1547 \times x \times \cos ( 38 )$$

Solve for x

$$x = \frac{\sqrt{2034276708458336872876809907095224089} + 4549580085948409467}{5000000000000000000} = 1\frac{9.7586081163264 \times 10^{17}}{5 \times 10^{18}} \approx 1.195172162$$

$$x=\frac{4549580085948409467-\sqrt{2034276708458336872876809907095224089}}{5000000000000000000}\approx 0.624659872$$

Steps Using the Quadratic Formula

Evaluate trigonometric functions in the problem

$${(0.766)} ^ {2} = {(x)} ^ {2} + {(1.1547)} ^ {2} - 2 \cdot 1.1547 \cdot x \cdot 0.788010753606722$$

Calculate 0.766 to the power of 2 and get 0.586756.

$$0.586756=x^{2}+1.1547^{2}-2\times 1.1547x\times 0.788010753606722$$

Calculate 1.1547 to the power of 2 and get 1.33333209.

$$0.586756=x^{2}+1.33333209-2\times 1.1547x\times 0.788010753606722$$

Multiply 2 and 1.1547 to get 2.3094.

$$0.586756=x^{2}+1.33333209-2.3094x\times 0.788010753606722$$

Multiply 2.3094 and 0.788010753606722 to get 1.8198320343793637868.

$$0.586756=x^{2}+1.33333209-1.8198320343793637868x$$

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

$$x^{2}+1.33333209-1.8198320343793637868x=0.586756$$

Subtract 0.586756 from both sides.

$$x^{2}+1.33333209-1.8198320343793637868x-0.586756=0$$

Subtract 0.586756 from 1.33333209 to get 0.74657609.

$$x^{2}+0.74657609-1.8198320343793637868x=0$$

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.

$$x^{2}-1.8198320343793637868x+0.74657609=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1.8198320343793637868 for b, and 0.74657609 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1.8198320343793637868\right)±\sqrt{\left(-1.8198320343793637868\right)^{2}-4\times 0.74657609}}{2}$$

Square -1.8198320343793637868 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-1.8198320343793637868\right)±\sqrt{3.31178863335333389966028958513523585424-4\times 0.74657609}}{2}$$

Multiply -4 times 0.74657609.

$$x=\frac{-\left(-1.8198320343793637868\right)±\sqrt{3.31178863335333389966028958513523585424-2.98630436}}{2}$$

Add 3.31178863335333389966028958513523585424 to -2.98630436 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-1.8198320343793637868\right)±\sqrt{0.32548427335333389966028958513523585424}}{2}$$

Take the square root of 0.32548427335333389966028958513523585424.

$$x=\frac{-\left(-1.8198320343793637868\right)±\frac{\sqrt{2034276708458336872876809907095224089}}{2500000000000000000}}{2}$$

The opposite of -1.8198320343793637868 is 1.8198320343793637868.

$$x=\frac{1.8198320343793637868±\frac{\sqrt{2034276708458336872876809907095224089}}{2500000000000000000}}{2}$$

Now solve the equation x=\frac{1.8198320343793637868±\frac{\sqrt{2034276708458336872876809907095224089}}{2500000000000000000}}{2} when ± is plus. Add 1.8198320343793637868 to \frac{\sqrt{2034276708458336872876809907095224089}}{2500000000000000000}.

$$x=\frac{\sqrt{2034276708458336872876809907095224089}+4549580085948409467}{2\times 2500000000000000000}$$

Divide \frac{4549580085948409467+\sqrt{2034276708458336872876809907095224089}}{2500000000000000000} by 2.

$$x=\frac{\sqrt{2034276708458336872876809907095224089}+4549580085948409467}{5000000000000000000}$$

Now solve the equation x=\frac{1.8198320343793637868±\frac{\sqrt{2034276708458336872876809907095224089}}{2500000000000000000}}{2} when ± is minus. Subtract \frac{\sqrt{2034276708458336872876809907095224089}}{2500000000000000000} from 1.8198320343793637868.

$$x=\frac{4549580085948409467-\sqrt{2034276708458336872876809907095224089}}{2\times 2500000000000000000}$$

Divide \frac{4549580085948409467-\sqrt{2034276708458336872876809907095224089}}{2500000000000000000} by 2.

$$x=\frac{4549580085948409467-\sqrt{2034276708458336872876809907095224089}}{5000000000000000000}$$

The equation is now solved.

$$x=\frac{\sqrt{2034276708458336872876809907095224089}+4549580085948409467}{5000000000000000000} x=\frac{4549580085948409467-\sqrt{2034276708458336872876809907095224089}}{5000000000000000000}$$

Steps for Completing the Square

Evaluate trigonometric functions in the problem

$${(0.766)} ^ {2} = {(x)} ^ {2} + {(1.1547)} ^ {2} - 2 \cdot 1.1547 \cdot x \cdot 0.788010753606722$$

Calculate 0.766 to the power of 2 and get 0.586756.

$$0.586756=x^{2}+1.1547^{2}-2\times 1.1547x\times 0.788010753606722$$

Calculate 1.1547 to the power of 2 and get 1.33333209.

$$0.586756=x^{2}+1.33333209-2\times 1.1547x\times 0.788010753606722$$

Multiply 2 and 1.1547 to get 2.3094.

$$0.586756=x^{2}+1.33333209-2.3094x\times 0.788010753606722$$

Multiply 2.3094 and 0.788010753606722 to get 1.8198320343793637868.

$$0.586756=x^{2}+1.33333209-1.8198320343793637868x$$

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

$$x^{2}+1.33333209-1.8198320343793637868x=0.586756$$

Subtract 1.33333209 from both sides.

$$x^{2}-1.8198320343793637868x=0.586756-1.33333209$$

Subtract 1.33333209 from 0.586756 to get -0.74657609.

$$x^{2}-1.8198320343793637868x=-0.74657609$$

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.

$$x^{2}-1.8198320343793637868x=-\frac{74657609}{100000000}$$

Divide -1.8198320343793637868, the coefficient of the x term, by 2 to get -0.9099160171896818934. Then add the square of -0.9099160171896818934 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1.8198320343793637868x+\left(-0.9099160171896818934\right)^{2}=-\frac{74657609}{100000000}+\left(-0.9099160171896818934\right)^{2}$$

Square -0.9099160171896818934 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-1.8198320343793637868x+0.82794715833833347491507239628380896356=-\frac{74657609}{100000000}+0.82794715833833347491507239628380896356$$

Add -\frac{74657609}{100000000} to 0.82794715833833347491507239628380896356 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-1.8198320343793637868x+0.82794715833833347491507239628380896356=\frac{2034276708458336872876809907095224089}{25000000000000000000000000000000000000}$$

Factor x^{2}-1.8198320343793637868x+0.82794715833833347491507239628380896356. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.9099160171896818934\right)^{2}=\frac{2034276708458336872876809907095224089}{25000000000000000000000000000000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.9099160171896818934\right)^{2}}=\sqrt{\frac{2034276708458336872876809907095224089}{25000000000000000000000000000000000000}}$$

Simplify.

$$x-0.9099160171896818934=\frac{\sqrt{2034276708458336872876809907095224089}}{5000000000000000000} x-0.9099160171896818934=-\frac{\sqrt{2034276708458336872876809907095224089}}{5000000000000000000}$$

Add 0.9099160171896818934 to both sides of the equation.

$$x=\frac{\sqrt{2034276708458336872876809907095224089}+4549580085948409467}{5000000000000000000} x=\frac{4549580085948409467-\sqrt{2034276708458336872876809907095224089}}{5000000000000000000}$$