Solve for q

q=\frac{1+\sqrt{3}i}{2}\approx 0.5+0.866025404i

q=\frac{-\sqrt{3}i+1}{2}\approx 0.5-0.866025404i

Quiz

Complex Number q ^ { 2 } - q + 1 = 0## Similar Problems from Web Search

https://socratic.org/questions/how-do-you-solve-q-2-5q-2-0

\displaystyle{q}=\frac{{5}}{{2}}\pm\frac{\sqrt{{{17}}}}{{2}} Explanation:Complete the square, then use the difference of squares identity: \displaystyle{a}^{{2}}-{b}^{{2}}={\left({a}-{b}\right)}{\left({a}+{b}\right)} ...

## Share

q^{2}-q+1=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.

q=\frac{-\left(-1\right)±\sqrt{1-4}}{2}

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

q=\frac{-\left(-1\right)±\sqrt{-3}}{2}

Add 1 to -4.

q=\frac{-\left(-1\right)±\sqrt{3}i}{2}

Take the square root of -3.

q=\frac{1±\sqrt{3}i}{2}

The opposite of -1 is 1.

q=\frac{1+\sqrt{3}i}{2}

Now solve the equation q=\frac{1±\sqrt{3}i}{2} when ± is plus. Add 1 to i\sqrt{3}.

q=\frac{-\sqrt{3}i+1}{2}

Now solve the equation q=\frac{1±\sqrt{3}i}{2} when ± is minus. Subtract i\sqrt{3} from 1.

q=\frac{1+\sqrt{3}i}{2} q=\frac{-\sqrt{3}i+1}{2}

The equation is now solved.

q^{2}-q+1=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.

q^{2}-q+1-1=-1

Subtract 1 from both sides of the equation.

q^{2}-q=-1

Subtracting 1 from itself leaves 0.

q^{2}-q+\left(-\frac{1}{2}\right)^{2}=-1+\left(-\frac{1}{2}\right)^{2}

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

q^{2}-q+\frac{1}{4}=-1+\frac{1}{4}

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

q^{2}-q+\frac{1}{4}=-\frac{3}{4}

Add -1 to \frac{1}{4}.

\left(q-\frac{1}{2}\right)^{2}=-\frac{3}{4}

Factor q^{2}-q+\frac{1}{4}. 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(q-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{3}{4}}

Take the square root of both sides of the equation.

q-\frac{1}{2}=\frac{\sqrt{3}i}{2} q-\frac{1}{2}=-\frac{\sqrt{3}i}{2}

Simplify.

q=\frac{1+\sqrt{3}i}{2} q=\frac{-\sqrt{3}i+1}{2}

Add \frac{1}{2} to both sides of the equation.

x ^ 2 -1x +1 = 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 = 1 rs = 1

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 = \frac{1}{2} - u s = \frac{1}{2} + u

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

(\frac{1}{2} - u) (\frac{1}{2} + u) = 1

To solve for unknown quantity u, substitute these in the product equation rs = 1

\frac{1}{4} - u^2 = 1

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

-u^2 = 1-\frac{1}{4} = \frac{3}{4}

Simplify the expression by subtracting \frac{1}{4} on both sides

u^2 = -\frac{3}{4} u = \pm\sqrt{-\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}i

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

r =\frac{1}{2} - \frac{\sqrt{3}}{2}i = 0.500 - 0.866i s = \frac{1}{2} + \frac{\sqrt{3}}{2}i = 0.500 + 0.866i

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