Solve -x^2-44x+65

$$- x ^ { 2 } - 44 x + 65$$

Factor

$$-\left(x-\left(-3\sqrt{61}-22\right)\right)\left(x-\left(3\sqrt{61}-22\right)\right)$$

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.

$$-x^{2}-44x+65=0$$

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.

$$x=\frac{-\left(-44\right)±\sqrt{\left(-44\right)^{2}-4\left(-1\right)\times 65}}{2\left(-1\right)}$$

Square -44.

$$x=\frac{-\left(-44\right)±\sqrt{1936-4\left(-1\right)\times 65}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-44\right)±\sqrt{1936+4\times 65}}{2\left(-1\right)}$$

Multiply 4 times 65.

$$x=\frac{-\left(-44\right)±\sqrt{1936+260}}{2\left(-1\right)}$$

Add 1936 to 260.

$$x=\frac{-\left(-44\right)±\sqrt{2196}}{2\left(-1\right)}$$

Take the square root of 2196.

$$x=\frac{-\left(-44\right)±6\sqrt{61}}{2\left(-1\right)}$$

The opposite of -44 is 44.

$$x=\frac{44±6\sqrt{61}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{44±6\sqrt{61}}{-2}$$

Now solve the equation x=\frac{44±6\sqrt{61}}{-2} when ± is plus. Add 44 to 6\sqrt{61}.

$$x=\frac{6\sqrt{61}+44}{-2}$$

Divide 44+6\sqrt{61} by -2.

$$x=-3\sqrt{61}-22$$

Now solve the equation x=\frac{44±6\sqrt{61}}{-2} when ± is minus. Subtract 6\sqrt{61} from 44.

$$x=\frac{44-6\sqrt{61}}{-2}$$

Divide 44-6\sqrt{61} by -2.

$$x=3\sqrt{61}-22$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -22-3\sqrt{61} for x_{1} and -22+3\sqrt{61} for x_{2}.

$$-x^{2}-44x+65=-\left(x-\left(-3\sqrt{61}-22\right)\right)\left(x-\left(3\sqrt{61}-22\right)\right)$$

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.

$$x ^ 2 +44x -65 = 0$$

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

$$r + s = -44 rs = -65$$

Two numbers r and s sum up to -44 exactly when the average of the two numbers is \frac{1}{2}*-44 = -22. 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>

$$r = -22 - u s = -22 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -65

$$(-22 - u) (-22 + u) = -65$$

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

$$484 - u^2 = -65$$

Simplify the expression by subtracting 484 on both sides

$$-u^2 = -65-484 = -549$$

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

$$u^2 = 549 u = \pm\sqrt{549} = \pm \sqrt{549}$$

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

$$r =-22 - \sqrt{549} = -45.431 s = -22 + \sqrt{549} = 1.431$$

Evaluate

$$65-44x-x^{2}$$