Solve x^2+23x-1

$$x ^ { 2 } + 23 x - 1$$

Factor

$$\left(x-\frac{-\sqrt{533}-23}{2}\right)\left(x-\frac{\sqrt{533}-23}{2}\right)$$

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.

$$x^{2}+23x-1=0$$

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.

$$x=\frac{-23±\sqrt{23^{2}-4\left(-1\right)}}{2}$$

Square 23.

$$x=\frac{-23±\sqrt{529-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$x=\frac{-23±\sqrt{529+4}}{2}$$

Add 529 to 4.

$$x=\frac{-23±\sqrt{533}}{2}$$

Now solve the equation x=\frac{-23±\sqrt{533}}{2} when ± is plus. Add -23 to \sqrt{533}.

$$x=\frac{\sqrt{533}-23}{2}$$

Now solve the equation x=\frac{-23±\sqrt{533}}{2} when ± is minus. Subtract \sqrt{533} from -23.

$$x=\frac{-\sqrt{533}-23}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-23+\sqrt{533}}{2} for x_{1} and \frac{-23-\sqrt{533}}{2} for x_{2}.

$$x^{2}+23x-1=\left(x-\frac{\sqrt{533}-23}{2}\right)\left(x-\frac{-\sqrt{533}-23}{2}\right)$$

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.

$$x ^ 2 +23x -1 = 0$$

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

$$r + s = -23 rs = -1$$

Two numbers r and s sum up to -23 exactly when the average of the two numbers is \frac{1}{2}*-23 = -\frac{23}{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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{23}{2} - u s = -\frac{23}{2} + u$$

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

$$(-\frac{23}{2} - u) (-\frac{23}{2} + u) = -1$$

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

$$\frac{529}{4} - u^2 = -1$$

Simplify the expression by subtracting \frac{529}{4} on both sides

$$-u^2 = -1-\frac{529}{4} = -\frac{533}{4}$$

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

$$u^2 = \frac{533}{4} u = \pm\sqrt{\frac{533}{4}} = \pm \frac{\sqrt{533}}{2}$$

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

$$r =-\frac{23}{2} - \frac{\sqrt{533}}{2} = -23.043 s = -\frac{23}{2} + \frac{\sqrt{533}}{2} = 0.043$$

Evaluate

$$x^{2}+23x-1$$