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4x^{2}+x-100=6
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.
4x^{2}+x-100-6=6-6
Subtract 6 from both sides of the equation.
4x^{2}+x-100-6=0
Subtracting 6 from itself leaves 0.
4x^{2}+x-106=0
Subtract 6 from -100.
x=\frac{-1±\sqrt{1^{2}-4\times 4\left(-106\right)}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 1 for b, and -106 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\times 4\left(-106\right)}}{2\times 4}
Square 1.
x=\frac{-1±\sqrt{1-16\left(-106\right)}}{2\times 4}
Multiply -4 times 4.
x=\frac{-1±\sqrt{1+1696}}{2\times 4}
Multiply -16 times -106.
x=\frac{-1±\sqrt{1697}}{2\times 4}
Add 1 to 1696.
x=\frac{-1±\sqrt{1697}}{8}
Multiply 2 times 4.
x=\frac{\sqrt{1697}-1}{8}
Now solve the equation x=\frac{-1±\sqrt{1697}}{8} when ± is plus. Add -1 to \sqrt{1697}.
x=\frac{-\sqrt{1697}-1}{8}
Now solve the equation x=\frac{-1±\sqrt{1697}}{8} when ± is minus. Subtract \sqrt{1697} from -1.
x=\frac{\sqrt{1697}-1}{8} x=\frac{-\sqrt{1697}-1}{8}
The equation is now solved.
4x^{2}+x-100=6
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}+x-100-\left(-100\right)=6-\left(-100\right)
Add 100 to both sides of the equation.
4x^{2}+x=6-\left(-100\right)
Subtracting -100 from itself leaves 0.
4x^{2}+x=106
Subtract -100 from 6.
\frac{4x^{2}+x}{4}=\frac{106}{4}
Divide both sides by 4.
x^{2}+\frac{1}{4}x=\frac{106}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}+\frac{1}{4}x=\frac{53}{2}
Reduce the fraction \frac{106}{4} to lowest terms by extracting and canceling out 2.
x^{2}+\frac{1}{4}x+\left(\frac{1}{8}\right)^{2}=\frac{53}{2}+\left(\frac{1}{8}\right)^{2}
Divide \frac{1}{4}, the coefficient of the x term, by 2 to get \frac{1}{8}. Then add the square of \frac{1}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{1}{4}x+\frac{1}{64}=\frac{53}{2}+\frac{1}{64}
Square \frac{1}{8} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{1}{4}x+\frac{1}{64}=\frac{1697}{64}
Add \frac{53}{2} to \frac{1}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{1}{8}\right)^{2}=\frac{1697}{64}
Factor x^{2}+\frac{1}{4}x+\frac{1}{64}. 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{1}{8}\right)^{2}}=\sqrt{\frac{1697}{64}}
Take the square root of both sides of the equation.
x+\frac{1}{8}=\frac{\sqrt{1697}}{8} x+\frac{1}{8}=-\frac{\sqrt{1697}}{8}
Simplify.
x=\frac{\sqrt{1697}-1}{8} x=\frac{-\sqrt{1697}-1}{8}
Subtract \frac{1}{8} from both sides of the equation.