Solve {r}{2i^2+36i}{+42}

$$\left. \begin{array} { r } { 2 i ^ { 2 } + 36 i } \\ { + 42 } \end{array} \right.$$

Evaluate

$$40+36i$$

Solution Steps

Calculate i to the power of 2 and get -1.

$$2\left(-1\right)+36i+42$$

Multiply 2 and -1 to get -2.

$$-2+36i+42$$

Do the additions.

$$40+36i$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 2

$$x ^ 2 +18x +21 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -18 rs = 21$$

Two numbers r and s sum up to -18 exactly when the average of the two numbers is \frac{1}{2}*-18 = -9. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -9 - u s = -9 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 21

$$(-9 - u) (-9 + u) = 21$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$81 - u^2 = 21$$

Simplify the expression by subtracting 81 on both sides

$$-u^2 = 21-81 = -60$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 60 u = \pm\sqrt{60} = \pm \sqrt{60}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-9 - \sqrt{60} = -16.746 s = -9 + \sqrt{60} = -1.254$$

Real Part

$$40$$

Solution Steps

Calculate i to the power of 2 and get -1.

$$Re(2\left(-1\right)+36i+42)$$

Multiply 2 and -1 to get -2.

$$Re(-2+36i+42)$$

Do the additions in -2+36i+42.

$$Re(40+36i)$$

The real part of 40+36i is 40.

$$40$$