Solve for x
x=\frac{2\sqrt{22}-5}{9}\approx 0.486759058
x=\frac{-2\sqrt{22}-5}{9}\approx -1.597870169
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9x^{2}+10x=7
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.
9x^{2}+10x-7=7-7
Subtract 7 from both sides of the equation.
9x^{2}+10x-7=0
Subtracting 7 from itself leaves 0.
x=\frac{-10±\sqrt{10^{2}-4\times 9\left(-7\right)}}{2\times 9}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 10 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-10±\sqrt{100-4\times 9\left(-7\right)}}{2\times 9}
Square 10.
x=\frac{-10±\sqrt{100-36\left(-7\right)}}{2\times 9}
Multiply -4 times 9.
x=\frac{-10±\sqrt{100+252}}{2\times 9}
Multiply -36 times -7.
x=\frac{-10±\sqrt{352}}{2\times 9}
Add 100 to 252.
x=\frac{-10±4\sqrt{22}}{2\times 9}
Take the square root of 352.
x=\frac{-10±4\sqrt{22}}{18}
Multiply 2 times 9.
x=\frac{4\sqrt{22}-10}{18}
Now solve the equation x=\frac{-10±4\sqrt{22}}{18} when ± is plus. Add -10 to 4\sqrt{22}.
x=\frac{2\sqrt{22}-5}{9}
Divide -10+4\sqrt{22} by 18.
x=\frac{-4\sqrt{22}-10}{18}
Now solve the equation x=\frac{-10±4\sqrt{22}}{18} when ± is minus. Subtract 4\sqrt{22} from -10.
x=\frac{-2\sqrt{22}-5}{9}
Divide -10-4\sqrt{22} by 18.
x=\frac{2\sqrt{22}-5}{9} x=\frac{-2\sqrt{22}-5}{9}
The equation is now solved.
9x^{2}+10x=7
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{9x^{2}+10x}{9}=\frac{7}{9}
Divide both sides by 9.
x^{2}+\frac{10}{9}x=\frac{7}{9}
Dividing by 9 undoes the multiplication by 9.
x^{2}+\frac{10}{9}x+\left(\frac{5}{9}\right)^{2}=\frac{7}{9}+\left(\frac{5}{9}\right)^{2}
Divide \frac{10}{9}, the coefficient of the x term, by 2 to get \frac{5}{9}. Then add the square of \frac{5}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{10}{9}x+\frac{25}{81}=\frac{7}{9}+\frac{25}{81}
Square \frac{5}{9} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{10}{9}x+\frac{25}{81}=\frac{88}{81}
Add \frac{7}{9} to \frac{25}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{5}{9}\right)^{2}=\frac{88}{81}
Factor x^{2}+\frac{10}{9}x+\frac{25}{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+\frac{5}{9}\right)^{2}}=\sqrt{\frac{88}{81}}
Take the square root of both sides of the equation.
x+\frac{5}{9}=\frac{2\sqrt{22}}{9} x+\frac{5}{9}=-\frac{2\sqrt{22}}{9}
Simplify.
x=\frac{2\sqrt{22}-5}{9} x=\frac{-2\sqrt{22}-5}{9}
Subtract \frac{5}{9} 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}