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\frac{1}{4}x^{2}-9x=28
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.
\frac{1}{4}x^{2}-9x-28=28-28
Subtract 28 from both sides of the equation.
\frac{1}{4}x^{2}-9x-28=0
Subtracting 28 from itself leaves 0.
x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times \frac{1}{4}\left(-28\right)}}{2\times \frac{1}{4}}
This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{4} for a, -9 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-9\right)±\sqrt{81-4\times \frac{1}{4}\left(-28\right)}}{2\times \frac{1}{4}}
Square -9.
x=\frac{-\left(-9\right)±\sqrt{81-\left(-28\right)}}{2\times \frac{1}{4}}
Multiply -4 times \frac{1}{4}.
x=\frac{-\left(-9\right)±\sqrt{81+28}}{2\times \frac{1}{4}}
Multiply -1 times -28.
x=\frac{-\left(-9\right)±\sqrt{109}}{2\times \frac{1}{4}}
Add 81 to 28.
x=\frac{9±\sqrt{109}}{2\times \frac{1}{4}}
The opposite of -9 is 9.
x=\frac{9±\sqrt{109}}{\frac{1}{2}}
Multiply 2 times \frac{1}{4}.
x=\frac{\sqrt{109}+9}{\frac{1}{2}}
Now solve the equation x=\frac{9±\sqrt{109}}{\frac{1}{2}} when ± is plus. Add 9 to \sqrt{109}.
x=2\sqrt{109}+18
Divide 9+\sqrt{109} by \frac{1}{2} by multiplying 9+\sqrt{109} by the reciprocal of \frac{1}{2}.
x=\frac{9-\sqrt{109}}{\frac{1}{2}}
Now solve the equation x=\frac{9±\sqrt{109}}{\frac{1}{2}} when ± is minus. Subtract \sqrt{109} from 9.
x=18-2\sqrt{109}
Divide 9-\sqrt{109} by \frac{1}{2} by multiplying 9-\sqrt{109} by the reciprocal of \frac{1}{2}.
x=2\sqrt{109}+18 x=18-2\sqrt{109}
The equation is now solved.
\frac{1}{4}x^{2}-9x=28
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{\frac{1}{4}x^{2}-9x}{\frac{1}{4}}=\frac{28}{\frac{1}{4}}
Multiply both sides by 4.
x^{2}+\left(-\frac{9}{\frac{1}{4}}\right)x=\frac{28}{\frac{1}{4}}
Dividing by \frac{1}{4} undoes the multiplication by \frac{1}{4}.
x^{2}-36x=\frac{28}{\frac{1}{4}}
Divide -9 by \frac{1}{4} by multiplying -9 by the reciprocal of \frac{1}{4}.
x^{2}-36x=112
Divide 28 by \frac{1}{4} by multiplying 28 by the reciprocal of \frac{1}{4}.
x^{2}-36x+\left(-18\right)^{2}=112+\left(-18\right)^{2}
Divide -36, the coefficient of the x term, by 2 to get -18. Then add the square of -18 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-36x+324=112+324
Square -18.
x^{2}-36x+324=436
Add 112 to 324.
\left(x-18\right)^{2}=436
Factor x^{2}-36x+324. 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-18\right)^{2}}=\sqrt{436}
Take the square root of both sides of the equation.
x-18=2\sqrt{109} x-18=-2\sqrt{109}
Simplify.
x=2\sqrt{109}+18 x=18-2\sqrt{109}
Add 18 to both sides of the equation.