Solve z^2+2z+4=0 | Microsoft Math Solver (2024)

z^{2}+2z+4=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.

z=\frac{-2±\sqrt{2^{2}-4\times 4}}{2}

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

z=\frac{-2±\sqrt{4-4\times 4}}{2}

Square 2.

z=\frac{-2±\sqrt{4-16}}{2}

Multiply -4 times 4.

z=\frac{-2±\sqrt{-12}}{2}

Add 4 to -16.

z=\frac{-2±2\sqrt{3}i}{2}

Take the square root of -12.

z=\frac{-2+2\sqrt{3}i}{2}

Now solve the equation z=\frac{-2±2\sqrt{3}i}{2} when ± is plus. Add -2 to 2i\sqrt{3}.

z=-1+\sqrt{3}i

Divide -2+2i\sqrt{3} by 2.

z=\frac{-2\sqrt{3}i-2}{2}

Now solve the equation z=\frac{-2±2\sqrt{3}i}{2} when ± is minus. Subtract 2i\sqrt{3} from -2.

z=-\sqrt{3}i-1

Divide -2-2i\sqrt{3} by 2.

z=-1+\sqrt{3}i z=-\sqrt{3}i-1

The equation is now solved.

z^{2}+2z+4=0

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.

z^{2}+2z+4-4=-4

Subtract 4 from both sides of the equation.

z^{2}+2z=-4

Subtracting 4 from itself leaves 0.

z^{2}+2z+1^{2}=-4+1^{2}

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

z^{2}+2z+1=-4+1

Square 1.

z^{2}+2z+1=-3

Add -4 to 1.

\left(z+1\right)^{2}=-3

Factor z^{2}+2z+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

\sqrt{\left(z+1\right)^{2}}=\sqrt{-3}

Take the square root of both sides of the equation.

z+1=\sqrt{3}i z+1=-\sqrt{3}i

Simplify.

z=-1+\sqrt{3}i z=-\sqrt{3}i-1

Subtract 1 from both sides of the equation.

x ^ 2 +2x +4 = 0

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.

r + s = -2 rs = 4

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 = -1 - u s = -1 + u

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

(-1 - u) (-1 + u) = 4

To solve for unknown quantity u, substitute these in the product equation rs = 4

1 - u^2 = 4

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

-u^2 = 4-1 = 3

Simplify the expression by subtracting 1 on both sides

u^2 = -3 u = \pm\sqrt{-3} = \pm \sqrt{3}i

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

r =-1 - \sqrt{3}i s = -1 + \sqrt{3}i

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