Solve 4i^2+12i+9=

$$4 i ^ { 2 } + 12 i + 9 =$$

Evaluate

$$5+12i$$

Solution Steps

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

$$4\left(-1\right)+12i+9$$

Multiply 4 and -1 to get -4.

$$-4+12i+9$$

Do the additions.

$$5+12i$$

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 4

$$x ^ 2 +3x +\frac{9}{4} = 0$$

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

$$r + s = -3 rs = \frac{9}{4}$$

Two numbers r and s sum up to -3 exactly when the average of the two numbers is \frac{1}{2}*-3 = -\frac{3}{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>

$$r = -\frac{3}{2} - u s = -\frac{3}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{9}{4}

$$(-\frac{3}{2} - u) (-\frac{3}{2} + u) = \frac{9}{4}$$

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

$$\frac{9}{4} - u^2 = \frac{9}{4}$$

Simplify the expression by subtracting \frac{9}{4} on both sides

$$-u^2 = \frac{9}{4}-\frac{9}{4} = 0$$

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

$$u^2 = 0 u = 0$$

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

$$r = s = -\frac{3}{2} = -1.500$$

Real Part

$$5$$

Solution Steps

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

$$Re(4\left(-1\right)+12i+9)$$

Multiply 4 and -1 to get -4.

$$Re(-4+12i+9)$$

Do the additions in -4+12i+9.

$$Re(5+12i)$$

The real part of 5+12i is 5.

$$5$$