Skip to main content
Solve for x
Tick mark Image
Graph

Similar Problems from Web Search

Share

x^{2}+x+6=180
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^{2}+x+6-180=180-180
Subtract 180 from both sides of the equation.
x^{2}+x+6-180=0
Subtracting 180 from itself leaves 0.
x^{2}+x-174=0
Subtract 180 from 6.
x=\frac{-1±\sqrt{1^{2}-4\left(-174\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -174 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\left(-174\right)}}{2}
Square 1.
x=\frac{-1±\sqrt{1+696}}{2}
Multiply -4 times -174.
x=\frac{-1±\sqrt{697}}{2}
Add 1 to 696.
x=\frac{\sqrt{697}-1}{2}
Now solve the equation x=\frac{-1±\sqrt{697}}{2} when ± is plus. Add -1 to \sqrt{697}.
x=\frac{-\sqrt{697}-1}{2}
Now solve the equation x=\frac{-1±\sqrt{697}}{2} when ± is minus. Subtract \sqrt{697} from -1.
x=\frac{\sqrt{697}-1}{2} x=\frac{-\sqrt{697}-1}{2}
The equation is now solved.
x^{2}+x+6=180
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}+x+6-6=180-6
Subtract 6 from both sides of the equation.
x^{2}+x=180-6
Subtracting 6 from itself leaves 0.
x^{2}+x=174
Subtract 6 from 180.
x^{2}+x+\left(\frac{1}{2}\right)^{2}=174+\left(\frac{1}{2}\right)^{2}
Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+x+\frac{1}{4}=174+\frac{1}{4}
Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+x+\frac{1}{4}=\frac{697}{4}
Add 174 to \frac{1}{4}.
\left(x+\frac{1}{2}\right)^{2}=\frac{697}{4}
Factor x^{2}+x+\frac{1}{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{1}{2}\right)^{2}}=\sqrt{\frac{697}{4}}
Take the square root of both sides of the equation.
x+\frac{1}{2}=\frac{\sqrt{697}}{2} x+\frac{1}{2}=-\frac{\sqrt{697}}{2}
Simplify.
x=\frac{\sqrt{697}-1}{2} x=\frac{-\sqrt{697}-1}{2}
Subtract \frac{1}{2} from both sides of the equation.