Solve for x
x=-12
x=4
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x^{2}+8x-48=0
Divide both sides by 3.
a+b=8 ab=1\left(-48\right)=-48
To solve the equation, factor the left hand side by grouping. First, left hand side 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=-4 b=12
The solution is the pair that gives sum 8.
\left(x^{2}-4x\right)+\left(12x-48\right)
Rewrite x^{2}+8x-48 as \left(x^{2}-4x\right)+\left(12x-48\right).
x\left(x-4\right)+12\left(x-4\right)
Factor out x in the first and 12 in the second group.
\left(x-4\right)\left(x+12\right)
Factor out common term x-4 by using distributive property.
x=4 x=-12
To find equation solutions, solve x-4=0 and x+12=0.
3x^{2}+24x-144=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.
x=\frac{-24±\sqrt{24^{2}-4\times 3\left(-144\right)}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 24 for b, and -144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-24±\sqrt{576-4\times 3\left(-144\right)}}{2\times 3}
Square 24.
x=\frac{-24±\sqrt{576-12\left(-144\right)}}{2\times 3}
Multiply -4 times 3.
x=\frac{-24±\sqrt{576+1728}}{2\times 3}
Multiply -12 times -144.
x=\frac{-24±\sqrt{2304}}{2\times 3}
Add 576 to 1728.
x=\frac{-24±48}{2\times 3}
Take the square root of 2304.
x=\frac{-24±48}{6}
Multiply 2 times 3.
x=\frac{24}{6}
Now solve the equation x=\frac{-24±48}{6} when ± is plus. Add -24 to 48.
x=4
Divide 24 by 6.
x=-\frac{72}{6}
Now solve the equation x=\frac{-24±48}{6} when ± is minus. Subtract 48 from -24.
x=-12
Divide -72 by 6.
x=4 x=-12
The equation is now solved.
3x^{2}+24x-144=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.
3x^{2}+24x-144-\left(-144\right)=-\left(-144\right)
Add 144 to both sides of the equation.
3x^{2}+24x=-\left(-144\right)
Subtracting -144 from itself leaves 0.
3x^{2}+24x=144
Subtract -144 from 0.
\frac{3x^{2}+24x}{3}=\frac{144}{3}
Divide both sides by 3.
x^{2}+\frac{24}{3}x=\frac{144}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+8x=\frac{144}{3}
Divide 24 by 3.
x^{2}+8x=48
Divide 144 by 3.
x^{2}+8x+4^{2}=48+4^{2}
Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+8x+16=48+16
Square 4.
x^{2}+8x+16=64
Add 48 to 16.
\left(x+4\right)^{2}=64
Factor x^{2}+8x+16. 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(x+4\right)^{2}}=\sqrt{64}
Take the square root of both sides of the equation.
x+4=8 x+4=-8
Simplify.
x=4 x=-12
Subtract 4 from both sides of the equation.
x ^ 2 +8x -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.This is achieved by dividing both sides of the equation by 3
r + s = -8 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 = -4 - u s = -4 + u
Two numbers r and s sum up to -8 exactly when the average of the two numbers is \frac{1}{2}*-8 = -4. 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>
(-4 - u) (-4 + u) = -48
To solve for unknown quantity u, substitute these in the product equation rs = -48
16 - u^2 = -48
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -48-16 = -64
Simplify the expression by subtracting 16 on both sides
u^2 = 64 u = \pm\sqrt{64} = \pm 8
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-4 - 8 = -12 s = -4 + 8 = 4
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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