Solve p^2+24p+23

$$p ^ { 2 } + 24 p + 23$$

Factor

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

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.

$$a+b=24 ab=1\times 23=23$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=23$$

Rewrite p^{2}+24p+23 as \left(p^{2}+p\right)+\left(23p+23\right).

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

Factor out p in the first and 23 in the second group.

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

Factor out common term p+1 by using distributive property.

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

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.

$$p^{2}+24p+23=0$$

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.

$$p=\frac{-24±\sqrt{24^{2}-4\times 23}}{2}$$

Square 24.

$$p=\frac{-24±\sqrt{576-4\times 23}}{2}$$

Multiply -4 times 23.

$$p=\frac{-24±\sqrt{576-92}}{2}$$

Add 576 to -92.

$$p=\frac{-24±\sqrt{484}}{2}$$

Take the square root of 484.

$$p=\frac{-24±22}{2}$$

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

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

Divide -2 by 2.

$$p=-1$$

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

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

Divide -46 by 2.

$$p=-23$$

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

$$p^{2}+24p+23=\left(p-\left(-1\right)\right)\left(p-\left(-23\right)\right)$$

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

$$p^{2}+24p+23=\left(p+1\right)\left(p+23\right)$$

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.

$$x ^ 2 +24x +23 = 0$$

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

$$r + s = -24 rs = 23$$

Two numbers r and s sum up to -24 exactly when the average of the two numbers is \frac{1}{2}*-24 = -12. 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>

$$r = -12 - u s = -12 + u$$

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

$$(-12 - u) (-12 + u) = 23$$

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

$$144 - u^2 = 23$$

Simplify the expression by subtracting 144 on both sides

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

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

$$u^2 = 121 u = \pm\sqrt{121} = \pm 11$$

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

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

Evaluate

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