Solve p^2-22p-23

$$p ^ { 2 } - 22 p - 23$$

Factor

$$\left(p-23\right)\left(p+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as p^{2}+ap+bp-23. To find a and b, set up a system to be solved.

$$a+b=-22 ab=1\left(-23\right)=-23$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-23 b=1$$

Rewrite p^{2}-22p-23 as \left(p^{2}-23p\right)+\left(p-23\right).

$$\left(p^{2}-23p\right)+\left(p-23\right)$$

Factor out p in p^{2}-23p.

$$p\left(p-23\right)+p-23$$

Factor out common term p-23 by using distributive property.

$$\left(p-23\right)\left(p+1\right)$$

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.

$$p^{2}-22p-23=0$$

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.

$$p=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\left(-23\right)}}{2}$$

Square -22.

$$p=\frac{-\left(-22\right)±\sqrt{484-4\left(-23\right)}}{2}$$

Multiply -4 times -23.

$$p=\frac{-\left(-22\right)±\sqrt{484+92}}{2}$$

Add 484 to 92.

$$p=\frac{-\left(-22\right)±\sqrt{576}}{2}$$

Take the square root of 576.

$$p=\frac{-\left(-22\right)±24}{2}$$

The opposite of -22 is 22.

$$p=\frac{22±24}{2}$$

Now solve the equation p=\frac{22±24}{2} when ± is plus. Add 22 to 24.

$$p=\frac{46}{2}$$

Divide 46 by 2.

$$p=23$$

Now solve the equation p=\frac{22±24}{2} when ± is minus. Subtract 24 from 22.

$$p=-\frac{2}{2}$$

Divide -2 by 2.

$$p=-1$$

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

$$p^{2}-22p-23=\left(p-23\right)\left(p-\left(-1\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$p^{2}-22p-23=\left(p-23\right)\left(p+1\right)$$

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.

$$x ^ 2 -22x -23 = 0$$

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

$$r + s = 22 rs = -23$$

Two numbers r and s sum up to 22 exactly when the average of the two numbers is \frac{1}{2}*22 = 11. 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>

$$r = 11 - u s = 11 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -23

$$(11 - u) (11 + u) = -23$$

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

$$121 - u^2 = -23$$

Simplify the expression by subtracting 121 on both sides

$$-u^2 = -23-121 = -144$$

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

$$u^2 = 144 u = \pm\sqrt{144} = \pm 12$$

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

$$r =11 - 12 = -1 s = 11 + 12 = 23$$

Evaluate

$$\left(p-23\right)\left(p+1\right)$$