Solve for x
x = \frac{\sqrt{465} + 9}{2} \approx 15.281929326
x=\frac{9-\sqrt{465}}{2}\approx -6.281929326
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38x=48x+192-2x\left(x-4\right)
Use the distributive property to multiply 48 by x+4.
38x+2x\left(x-4\right)=48x+192
Add 2x\left(x-4\right) to both sides.
38x+2x^{2}-8x=48x+192
Use the distributive property to multiply 2x by x-4.
30x+2x^{2}=48x+192
Combine 38x and -8x to get 30x.
30x+2x^{2}-48x=192
Subtract 48x from both sides.
-18x+2x^{2}=192
Combine 30x and -48x to get -18x.
-18x+2x^{2}-192=0
Subtract 192 from both sides.
2x^{2}-18x-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{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 2\left(-192\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -18 for b, and -192 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-18\right)±\sqrt{324-4\times 2\left(-192\right)}}{2\times 2}
Square -18.
x=\frac{-\left(-18\right)±\sqrt{324-8\left(-192\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-18\right)±\sqrt{324+1536}}{2\times 2}
Multiply -8 times -192.
x=\frac{-\left(-18\right)±\sqrt{1860}}{2\times 2}
Add 324 to 1536.
x=\frac{-\left(-18\right)±2\sqrt{465}}{2\times 2}
Take the square root of 1860.
x=\frac{18±2\sqrt{465}}{2\times 2}
The opposite of -18 is 18.
x=\frac{18±2\sqrt{465}}{4}
Multiply 2 times 2.
x=\frac{2\sqrt{465}+18}{4}
Now solve the equation x=\frac{18±2\sqrt{465}}{4} when ± is plus. Add 18 to 2\sqrt{465}.
x=\frac{\sqrt{465}+9}{2}
Divide 18+2\sqrt{465} by 4.
x=\frac{18-2\sqrt{465}}{4}
Now solve the equation x=\frac{18±2\sqrt{465}}{4} when ± is minus. Subtract 2\sqrt{465} from 18.
x=\frac{9-\sqrt{465}}{2}
Divide 18-2\sqrt{465} by 4.
x=\frac{\sqrt{465}+9}{2} x=\frac{9-\sqrt{465}}{2}
The equation is now solved.
38x=48x+192-2x\left(x-4\right)
Use the distributive property to multiply 48 by x+4.
38x+2x\left(x-4\right)=48x+192
Add 2x\left(x-4\right) to both sides.
38x+2x^{2}-8x=48x+192
Use the distributive property to multiply 2x by x-4.
30x+2x^{2}=48x+192
Combine 38x and -8x to get 30x.
30x+2x^{2}-48x=192
Subtract 48x from both sides.
-18x+2x^{2}=192
Combine 30x and -48x to get -18x.
2x^{2}-18x=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.
\frac{2x^{2}-18x}{2}=\frac{192}{2}
Divide both sides by 2.
x^{2}+\left(-\frac{18}{2}\right)x=\frac{192}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}-9x=\frac{192}{2}
Divide -18 by 2.
x^{2}-9x=96
Divide 192 by 2.
x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=96+\left(-\frac{9}{2}\right)^{2}
Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-9x+\frac{81}{4}=96+\frac{81}{4}
Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-9x+\frac{81}{4}=\frac{465}{4}
Add 96 to \frac{81}{4}.
\left(x-\frac{9}{2}\right)^{2}=\frac{465}{4}
Factor x^{2}-9x+\frac{81}{4}. 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-\frac{9}{2}\right)^{2}}=\sqrt{\frac{465}{4}}
Take the square root of both sides of the equation.
x-\frac{9}{2}=\frac{\sqrt{465}}{2} x-\frac{9}{2}=-\frac{\sqrt{465}}{2}
Simplify.
x=\frac{\sqrt{465}+9}{2} x=\frac{9-\sqrt{465}}{2}
Add \frac{9}{2} to both sides of the equation.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}