Solve i^2-7i+2

$$i ^ { 2 } - 7 i + 2$$

Evaluate

$$1-7i$$

Solution Steps

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

$$-1-7i+2$$

Do the additions.

$$1-7i$$

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.

$$x ^ 2 -7x +2 = 0$$

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

$$r + s = 7 rs = 2$$

Two numbers r and s sum up to 7 exactly when the average of the two numbers is \frac{1}{2}*7 = \frac{7}{2}. 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>

$$r = \frac{7}{2} - u s = \frac{7}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 2

$$(\frac{7}{2} - u) (\frac{7}{2} + u) = 2$$

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

$$\frac{49}{4} - u^2 = 2$$

Simplify the expression by subtracting \frac{49}{4} on both sides

$$-u^2 = 2-\frac{49}{4} = -\frac{41}{4}$$

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

$$u^2 = \frac{41}{4} u = \pm\sqrt{\frac{41}{4}} = \pm \frac{\sqrt{41}}{2}$$

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

$$r =\frac{7}{2} - \frac{\sqrt{41}}{2} = 0.298 s = \frac{7}{2} + \frac{\sqrt{41}}{2} = 6.702$$

Real Part

$$1$$

Solution Steps

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

$$Re(-1-7i+2)$$

Do the additions in -1-7i+2.

$$Re(1-7i)$$

The real part of 1-7i is 1.

$$1$$