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x\left(58-3x\right)=\left(x+4\right)\times 48+x\left(x+4\right)\left(-4\right)
Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+4\right), the least common multiple of 4+x,x.
58x-3x^{2}=\left(x+4\right)\times 48+x\left(x+4\right)\left(-4\right)
Use the distributive property to multiply x by 58-3x.
58x-3x^{2}=48x+192+x\left(x+4\right)\left(-4\right)
Use the distributive property to multiply x+4 by 48.
58x-3x^{2}=48x+192+\left(x^{2}+4x\right)\left(-4\right)
Use the distributive property to multiply x by x+4.
58x-3x^{2}=48x+192-4x^{2}-16x
Use the distributive property to multiply x^{2}+4x by -4.
58x-3x^{2}=32x+192-4x^{2}
Combine 48x and -16x to get 32x.
58x-3x^{2}-32x=192-4x^{2}
Subtract 32x from both sides.
26x-3x^{2}=192-4x^{2}
Combine 58x and -32x to get 26x.
26x-3x^{2}-192=-4x^{2}
Subtract 192 from both sides.
26x-3x^{2}-192+4x^{2}=0
Add 4x^{2} to both sides.
26x+x^{2}-192=0
Combine -3x^{2} and 4x^{2} to get x^{2}.
x^{2}+26x-192=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{-26±\sqrt{26^{2}-4\left(-192\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 26 for b, and -192 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-26±\sqrt{676-4\left(-192\right)}}{2}
Square 26.
x=\frac{-26±\sqrt{676+768}}{2}
Multiply -4 times -192.
x=\frac{-26±\sqrt{1444}}{2}
Add 676 to 768.
x=\frac{-26±38}{2}
Take the square root of 1444.
x=\frac{12}{2}
Now solve the equation x=\frac{-26±38}{2} when ± is plus. Add -26 to 38.
x=6
Divide 12 by 2.
x=-\frac{64}{2}
Now solve the equation x=\frac{-26±38}{2} when ± is minus. Subtract 38 from -26.
x=-32
Divide -64 by 2.
x=6 x=-32
The equation is now solved.
x\left(58-3x\right)=\left(x+4\right)\times 48+x\left(x+4\right)\left(-4\right)
Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+4\right), the least common multiple of 4+x,x.
58x-3x^{2}=\left(x+4\right)\times 48+x\left(x+4\right)\left(-4\right)
Use the distributive property to multiply x by 58-3x.
58x-3x^{2}=48x+192+x\left(x+4\right)\left(-4\right)
Use the distributive property to multiply x+4 by 48.
58x-3x^{2}=48x+192+\left(x^{2}+4x\right)\left(-4\right)
Use the distributive property to multiply x by x+4.
58x-3x^{2}=48x+192-4x^{2}-16x
Use the distributive property to multiply x^{2}+4x by -4.
58x-3x^{2}=32x+192-4x^{2}
Combine 48x and -16x to get 32x.
58x-3x^{2}-32x=192-4x^{2}
Subtract 32x from both sides.
26x-3x^{2}=192-4x^{2}
Combine 58x and -32x to get 26x.
26x-3x^{2}+4x^{2}=192
Add 4x^{2} to both sides.
26x+x^{2}=192
Combine -3x^{2} and 4x^{2} to get x^{2}.
x^{2}+26x=192
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.
x^{2}+26x+13^{2}=192+13^{2}
Divide 26, the coefficient of the x term, by 2 to get 13. Then add the square of 13 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+26x+169=192+169
Square 13.
x^{2}+26x+169=361
Add 192 to 169.
\left(x+13\right)^{2}=361
Factor x^{2}+26x+169. 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+13\right)^{2}}=\sqrt{361}
Take the square root of both sides of the equation.
x+13=19 x+13=-19
Simplify.
x=6 x=-32
Subtract 13 from both sides of the equation.