Solve for x
x=\frac{2\sqrt{106}-18}{5}\approx 0.518252056
x=\frac{-2\sqrt{106}-18}{5}\approx -7.718252056
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5x^{2}+36x-20=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{-36±\sqrt{36^{2}-4\times 5\left(-20\right)}}{2\times 5}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 36 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-36±\sqrt{1296-4\times 5\left(-20\right)}}{2\times 5}
Square 36.
x=\frac{-36±\sqrt{1296-20\left(-20\right)}}{2\times 5}
Multiply -4 times 5.
x=\frac{-36±\sqrt{1296+400}}{2\times 5}
Multiply -20 times -20.
x=\frac{-36±\sqrt{1696}}{2\times 5}
Add 1296 to 400.
x=\frac{-36±4\sqrt{106}}{2\times 5}
Take the square root of 1696.
x=\frac{-36±4\sqrt{106}}{10}
Multiply 2 times 5.
x=\frac{4\sqrt{106}-36}{10}
Now solve the equation x=\frac{-36±4\sqrt{106}}{10} when ± is plus. Add -36 to 4\sqrt{106}.
x=\frac{2\sqrt{106}-18}{5}
Divide -36+4\sqrt{106} by 10.
x=\frac{-4\sqrt{106}-36}{10}
Now solve the equation x=\frac{-36±4\sqrt{106}}{10} when ± is minus. Subtract 4\sqrt{106} from -36.
x=\frac{-2\sqrt{106}-18}{5}
Divide -36-4\sqrt{106} by 10.
x=\frac{2\sqrt{106}-18}{5} x=\frac{-2\sqrt{106}-18}{5}
The equation is now solved.
5x^{2}+36x-20=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.
5x^{2}+36x-20-\left(-20\right)=-\left(-20\right)
Add 20 to both sides of the equation.
5x^{2}+36x=-\left(-20\right)
Subtracting -20 from itself leaves 0.
5x^{2}+36x=20
Subtract -20 from 0.
\frac{5x^{2}+36x}{5}=\frac{20}{5}
Divide both sides by 5.
x^{2}+\frac{36}{5}x=\frac{20}{5}
Dividing by 5 undoes the multiplication by 5.
x^{2}+\frac{36}{5}x=4
Divide 20 by 5.
x^{2}+\frac{36}{5}x+\left(\frac{18}{5}\right)^{2}=4+\left(\frac{18}{5}\right)^{2}
Divide \frac{36}{5}, the coefficient of the x term, by 2 to get \frac{18}{5}. Then add the square of \frac{18}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{36}{5}x+\frac{324}{25}=4+\frac{324}{25}
Square \frac{18}{5} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{36}{5}x+\frac{324}{25}=\frac{424}{25}
Add 4 to \frac{324}{25}.
\left(x+\frac{18}{5}\right)^{2}=\frac{424}{25}
Factor x^{2}+\frac{36}{5}x+\frac{324}{25}. 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{18}{5}\right)^{2}}=\sqrt{\frac{424}{25}}
Take the square root of both sides of the equation.
x+\frac{18}{5}=\frac{2\sqrt{106}}{5} x+\frac{18}{5}=-\frac{2\sqrt{106}}{5}
Simplify.
x=\frac{2\sqrt{106}-18}{5} x=\frac{-2\sqrt{106}-18}{5}
Subtract \frac{18}{5} from both sides of the equation.
Examples
Quadratic equation
{ 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}