Solve for p
p=\sqrt{385}+19\approx 38.62141687
p=19-\sqrt{385}\approx -0.62141687
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p^{2}-38p-24=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(-38\right)±\sqrt{\left(-38\right)^{2}-4\left(-24\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -38 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
p=\frac{-\left(-38\right)±\sqrt{1444-4\left(-24\right)}}{2}
Square -38.
p=\frac{-\left(-38\right)±\sqrt{1444+96}}{2}
Multiply -4 times -24.
p=\frac{-\left(-38\right)±\sqrt{1540}}{2}
Add 1444 to 96.
p=\frac{-\left(-38\right)±2\sqrt{385}}{2}
Take the square root of 1540.
p=\frac{38±2\sqrt{385}}{2}
The opposite of -38 is 38.
p=\frac{2\sqrt{385}+38}{2}
Now solve the equation p=\frac{38±2\sqrt{385}}{2} when ± is plus. Add 38 to 2\sqrt{385}.
p=\sqrt{385}+19
Divide 38+2\sqrt{385} by 2.
p=\frac{38-2\sqrt{385}}{2}
Now solve the equation p=\frac{38±2\sqrt{385}}{2} when ± is minus. Subtract 2\sqrt{385} from 38.
p=19-\sqrt{385}
Divide 38-2\sqrt{385} by 2.
p=\sqrt{385}+19 p=19-\sqrt{385}
The equation is now solved.
p^{2}-38p-24=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
p^{2}-38p-24-\left(-24\right)=-\left(-24\right)
Add 24 to both sides of the equation.
p^{2}-38p=-\left(-24\right)
Subtracting -24 from itself leaves 0.
p^{2}-38p=24
Subtract -24 from 0.
p^{2}-38p+\left(-19\right)^{2}=24+\left(-19\right)^{2}
Divide -38, the coefficient of the x term, by 2 to get -19. Then add the square of -19 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
p^{2}-38p+361=24+361
Square -19.
p^{2}-38p+361=385
Add 24 to 361.
\left(p-19\right)^{2}=385
Factor p^{2}-38p+361. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(p-19\right)^{2}}=\sqrt{385}
Take the square root of both sides of the equation.
p-19=\sqrt{385} p-19=-\sqrt{385}
Simplify.
p=\sqrt{385}+19 p=19-\sqrt{385}
Add 19 to both sides of the equation.
x ^ 2 -38x -24 = 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 = 38 rs = -24
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 = 19 - u s = 19 + u
Two numbers r and s sum up to 38 exactly when the average of the two numbers is \frac{1}{2}*38 = 19. 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>
(19 - u) (19 + u) = -24
To solve for unknown quantity u, substitute these in the product equation rs = -24
361 - u^2 = -24
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -24-361 = -385
Simplify the expression by subtracting 361 on both sides
u^2 = 385 u = \pm\sqrt{385} = \pm \sqrt{385}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =19 - \sqrt{385} = -0.621 s = 19 + \sqrt{385} = 38.621
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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