Factor
\left(p+1\right)\left(p+23\right)
Evaluate
\left(p+1\right)\left(p+23\right)
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a+b=24 ab=1\times 23=23
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=1 b=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.
\left(p^{2}+p\right)+\left(23p+23\right)
Rewrite p^{2}+24p+23 as \left(p^{2}+p\right)+\left(23p+23\right).
p\left(p+1\right)+23\left(p+1\right)
Factor out p in the first and 23 in the second group.
\left(p+1\right)\left(p+23\right)
Factor out common term p+1 by using distributive property.
p^{2}+24p+23=0
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=\frac{-24±\sqrt{24^{2}-4\times 23}}{2}
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{576-4\times 23}}{2}
Square 24.
p=\frac{-24±\sqrt{576-92}}{2}
Multiply -4 times 23.
p=\frac{-24±\sqrt{484}}{2}
Add 576 to -92.
p=\frac{-24±22}{2}
Take the square root of 484.
p=-\frac{2}{2}
Now solve the equation p=\frac{-24±22}{2} when ± is plus. Add -24 to 22.
p=-1
Divide -2 by 2.
p=-\frac{46}{2}
Now solve the equation p=\frac{-24±22}{2} when ± is minus. Subtract 22 from -24.
p=-23
Divide -46 by 2.
p^{2}+24p+23=\left(p-\left(-1\right)\right)\left(p-\left(-23\right)\right)
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+1\right)\left(p+23\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 +24x +23 = 0
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.
r + s = -24 rs = 23
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 = -12 - u s = -12 + u
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>
(-12 - u) (-12 + u) = 23
To solve for unknown quantity u, substitute these in the product equation rs = 23
144 - u^2 = 23
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 23-144 = -121
Simplify the expression by subtracting 144 on both sides
u^2 = 121 u = \pm\sqrt{121} = \pm 11
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-12 - 11 = -23 s = -12 + 11 = -1
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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