a+b=2 ab=-24
To solve the equation, factor w^{2}+2w-24 using formula w^{2}+\left(a+b\right)w+ab=\left(w+a\right)\left(w+b\right). To find a and b, set up a system to be solved.
-1,24 -2,12 -3,8 -4,6
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -24.
-1+24=23 -2+12=10 -3+8=5 -4+6=2
Calculate the sum for each pair.
a=-4 b=6
The solution is the pair that gives sum 2.
\left(w-4\right)\left(w+6\right)
Rewrite factored expression \left(w+a\right)\left(w+b\right) using the obtained values.
w=4 w=-6
To find equation solutions, solve w-4=0 and w+6=0.
a+b=2 ab=1\left(-24\right)=-24
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as w^{2}+aw+bw-24. To find a and b, set up a system to be solved.
-1,24 -2,12 -3,8 -4,6
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -24.
-1+24=23 -2+12=10 -3+8=5 -4+6=2
Calculate the sum for each pair.
a=-4 b=6
The solution is the pair that gives sum 2.
\left(w^{2}-4w\right)+\left(6w-24\right)
Rewrite w^{2}+2w-24 as \left(w^{2}-4w\right)+\left(6w-24\right).
w\left(w-4\right)+6\left(w-4\right)
Factor out w in the first and 6 in the second group.
\left(w-4\right)\left(w+6\right)
Factor out common term w-4 by using distributive property.
w=4 w=-6
To find equation solutions, solve w-4=0 and w+6=0.
w^{2}+2w-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.
w=\frac{-2±\sqrt{2^{2}-4\left(-24\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
w=\frac{-2±\sqrt{4-4\left(-24\right)}}{2}
Square 2.
w=\frac{-2±\sqrt{4+96}}{2}
Multiply -4 times -24.
w=\frac{-2±\sqrt{100}}{2}
Add 4 to 96.
w=\frac{-2±10}{2}
Take the square root of 100.
w=\frac{8}{2}
Now solve the equation w=\frac{-2±10}{2} when ± is plus. Add -2 to 10.
w=4
Divide 8 by 2.
w=-\frac{12}{2}
Now solve the equation w=\frac{-2±10}{2} when ± is minus. Subtract 10 from -2.
w=-6
Divide -12 by 2.
w=4 w=-6
The equation is now solved.
w^{2}+2w-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.
w^{2}+2w-24-\left(-24\right)=-\left(-24\right)
Add 24 to both sides of the equation.
w^{2}+2w=-\left(-24\right)
Subtracting -24 from itself leaves 0.
w^{2}+2w=24
Subtract -24 from 0.
w^{2}+2w+1^{2}=24+1^{2}
Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
w^{2}+2w+1=24+1
Square 1.
w^{2}+2w+1=25
Add 24 to 1.
\left(w+1\right)^{2}=25
Factor w^{2}+2w+1. 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(w+1\right)^{2}}=\sqrt{25}
Take the square root of both sides of the equation.
w+1=5 w+1=-5
Simplify.
w=4 w=-6
Subtract 1 from both sides of the equation.
x ^ 2 +2x -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 = -2 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 = -1 - u s = -1 + u
Two numbers r and s sum up to -2 exactly when the average of the two numbers is \frac{1}{2}*-2 = -1. 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>
(-1 - u) (-1 + u) = -24
To solve for unknown quantity u, substitute these in the product equation rs = -24
1 - u^2 = -24
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -24-1 = -25
Simplify the expression by subtracting 1 on both sides
u^2 = 25 u = \pm\sqrt{25} = \pm 5
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-1 - 5 = -6 s = -1 + 5 = 4
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
