Factor
\left(x-6\right)\left(x+8\right)
Evaluate
\left(x-6\right)\left(x+8\right)
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a+b=2 ab=1\left(-48\right)=-48
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-48. To find a and b, set up a system to be solved.
-1,48 -2,24 -3,16 -4,12 -6,8
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 -48.
-1+48=47 -2+24=22 -3+16=13 -4+12=8 -6+8=2
Calculate the sum for each pair.
a=-6 b=8
The solution is the pair that gives sum 2.
\left(x^{2}-6x\right)+\left(8x-48\right)
Rewrite x^{2}+2x-48 as \left(x^{2}-6x\right)+\left(8x-48\right).
x\left(x-6\right)+8\left(x-6\right)
Factor out x in the first and 8 in the second group.
\left(x-6\right)\left(x+8\right)
Factor out common term x-6 by using distributive property.
x^{2}+2x-48=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.
x=\frac{-2±\sqrt{2^{2}-4\left(-48\right)}}{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.
x=\frac{-2±\sqrt{4-4\left(-48\right)}}{2}
Square 2.
x=\frac{-2±\sqrt{4+192}}{2}
Multiply -4 times -48.
x=\frac{-2±\sqrt{196}}{2}
Add 4 to 192.
x=\frac{-2±14}{2}
Take the square root of 196.
x=\frac{12}{2}
Now solve the equation x=\frac{-2±14}{2} when ± is plus. Add -2 to 14.
x=6
Divide 12 by 2.
x=-\frac{16}{2}
Now solve the equation x=\frac{-2±14}{2} when ± is minus. Subtract 14 from -2.
x=-8
Divide -16 by 2.
x^{2}+2x-48=\left(x-6\right)\left(x-\left(-8\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6 for x_{1} and -8 for x_{2}.
x^{2}+2x-48=\left(x-6\right)\left(x+8\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 +2x -48 = 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 = -48
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) = -48
To solve for unknown quantity u, substitute these in the product equation rs = -48
1 - u^2 = -48
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -48-1 = -49
Simplify the expression by subtracting 1 on both sides
u^2 = 49 u = \pm\sqrt{49} = \pm 7
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-1 - 7 = -8 s = -1 + 7 = 6
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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Simultaneous equation
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Integration
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Limits
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