Solve x^2-11x+12

$$x ^ { 2 } - 11 x + 12$$

Factor

$$\left(x-\frac{11-\sqrt{73}}{2}\right)\left(x-\frac{\sqrt{73}+11}{2}\right)$$

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.

$$x^{2}-11x+12=0$$

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.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 12}}{2}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 12}}{2}$$

Multiply -4 times 12.

$$x=\frac{-\left(-11\right)±\sqrt{121-48}}{2}$$

Add 121 to -48.

$$x=\frac{-\left(-11\right)±\sqrt{73}}{2}$$

The opposite of -11 is 11.

$$x=\frac{11±\sqrt{73}}{2}$$

Now solve the equation x=\frac{11±\sqrt{73}}{2} when ± is plus. Add 11 to \sqrt{73}.

$$x=\frac{\sqrt{73}+11}{2}$$

Now solve the equation x=\frac{11±\sqrt{73}}{2} when ± is minus. Subtract \sqrt{73} from 11.

$$x=\frac{11-\sqrt{73}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11+\sqrt{73}}{2} for x_{1} and \frac{11-\sqrt{73}}{2} for x_{2}.

$$x^{2}-11x+12=\left(x-\frac{\sqrt{73}+11}{2}\right)\left(x-\frac{11-\sqrt{73}}{2}\right)$$

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.

$$x ^ 2 -11x +12 = 0$$

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

$$r + s = 11 rs = 12$$

Two numbers r and s sum up to 11 exactly when the average of the two numbers is \frac{1}{2}*11 = \frac{11}{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>

$$r = \frac{11}{2} - u s = \frac{11}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 12

$$(\frac{11}{2} - u) (\frac{11}{2} + u) = 12$$

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

$$\frac{121}{4} - u^2 = 12$$

Simplify the expression by subtracting \frac{121}{4} on both sides

$$-u^2 = 12-\frac{121}{4} = -\frac{73}{4}$$

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

$$u^2 = \frac{73}{4} u = \pm\sqrt{\frac{73}{4}} = \pm \frac{\sqrt{73}}{2}$$

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

$$r =\frac{11}{2} - \frac{\sqrt{73}}{2} = 1.228 s = \frac{11}{2} + \frac{\sqrt{73}}{2} = 9.772$$

Evaluate

$$x^{2}-11x+12$$