Solve x^2-3x+1

$$x ^ { 2 } - 3 x + 1$$

Factor

$$\left(x-\frac{3-\sqrt{5}}{2}\right)\left(x-\frac{\sqrt{5}+3}{2}\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$x^{2}-3x+1=0$$

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.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4}}{2}$$

Add 9 to -4.

$$x=\frac{-\left(-3\right)±\sqrt{5}}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{5}}{2}$$

Now solve the equation x=\frac{3±\sqrt{5}}{2} when ± is plus. Add 3 to \sqrt{5}.

$$x=\frac{\sqrt{5}+3}{2}$$

Now solve the equation x=\frac{3±\sqrt{5}}{2} when ± is minus. Subtract \sqrt{5} from 3.

$$x=\frac{3-\sqrt{5}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+\sqrt{5}}{2} for x_{1} and \frac{3-\sqrt{5}}{2} for x_{2}.

$$x^{2}-3x+1=\left(x-\frac{\sqrt{5}+3}{2}\right)\left(x-\frac{3-\sqrt{5}}{2}\right)$$

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.

$$x ^ 2 -3x +1 = 0$$

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

$$r + s = 3 rs = 1$$

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 = 1

$$(\frac{3}{2} - u) (\frac{3}{2} + u) = 1$$

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

$$\frac{9}{4} - u^2 = 1$$

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

$$-u^2 = 1-\frac{9}{4} = -\frac{5}{4}$$

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

$$u^2 = \frac{5}{4} u = \pm\sqrt{\frac{5}{4}} = \pm \frac{\sqrt{5}}{2}$$

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

$$r =\frac{3}{2} - \frac{\sqrt{5}}{2} = 0.382 s = \frac{3}{2} + \frac{\sqrt{5}}{2} = 2.618$$

Evaluate

$$x^{2}-3x+1$$