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$4 \exponential{(x)}{2} - 72 x + 324 = 0 $
Solve for x
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4x^{2}-72x+324=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(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 4\times 324}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -72 for b, and 324 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-72\right)±\sqrt{5184-4\times 4\times 324}}{2\times 4}
Square -72.
x=\frac{-\left(-72\right)±\sqrt{5184-16\times 324}}{2\times 4}
Multiply -4 times 4.
x=\frac{-\left(-72\right)±\sqrt{5184-5184}}{2\times 4}
Multiply -16 times 324.
x=\frac{-\left(-72\right)±\sqrt{0}}{2\times 4}
Add 5184 to -5184.
x=-\frac{-72}{2\times 4}
Take the square root of 0.
x=\frac{72}{2\times 4}
The opposite of -72 is 72.
x=\frac{72}{8}
Multiply 2 times 4.
x=9
Divide 72 by 8.
4x^{2}-72x+324=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.
4x^{2}-72x+324-324=-324
Subtract 324 from both sides of the equation.
4x^{2}-72x=-324
Subtracting 324 from itself leaves 0.
\frac{4x^{2}-72x}{4}=\frac{-324}{4}
Divide both sides by 4.
x^{2}+\frac{-72}{4}x=\frac{-324}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}-18x=\frac{-324}{4}
Divide -72 by 4.
x^{2}-18x=-81
Divide -324 by 4.
x^{2}-18x+\left(-9\right)^{2}=-81+\left(-9\right)^{2}
Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-18x+81=-81+81
Square -9.
x^{2}-18x+81=0
Add -81 to 81.
\left(x-9\right)^{2}=0
Factor x^{2}-18x+81. 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-9\right)^{2}}=\sqrt{0}
Take the square root of both sides of the equation.
x-9=0 x-9=0
Simplify.
x=9 x=9
Add 9 to both sides of the equation.
x=9
The equation is now solved. Solutions are the same.