Solve for x
x=1
x=\frac{3}{5}=0.6
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5x^{2}-8x=-3
Subtract 8x from both sides.
5x^{2}-8x+3=0
Add 3 to both sides.
a+b=-8 ab=5\times 3=15
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 5x^{2}+ax+bx+3. To find a and b, set up a system to be solved.
-1,-15 -3,-5
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 15.
-1-15=-16 -3-5=-8
Calculate the sum for each pair.
a=-5 b=-3
The solution is the pair that gives sum -8.
\left(5x^{2}-5x\right)+\left(-3x+3\right)
Rewrite 5x^{2}-8x+3 as \left(5x^{2}-5x\right)+\left(-3x+3\right).
5x\left(x-1\right)-3\left(x-1\right)
Factor out 5x in the first and -3 in the second group.
\left(x-1\right)\left(5x-3\right)
Factor out common term x-1 by using distributive property.
x=1 x=\frac{3}{5}
To find equation solutions, solve x-1=0 and 5x-3=0.
5x^{2}-8x=-3
Subtract 8x from both sides.
5x^{2}-8x+3=0
Add 3 to both sides.
x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 5\times 3}}{2\times 5}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -8 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-8\right)±\sqrt{64-4\times 5\times 3}}{2\times 5}
Square -8.
x=\frac{-\left(-8\right)±\sqrt{64-20\times 3}}{2\times 5}
Multiply -4 times 5.
x=\frac{-\left(-8\right)±\sqrt{64-60}}{2\times 5}
Multiply -20 times 3.
x=\frac{-\left(-8\right)±\sqrt{4}}{2\times 5}
Add 64 to -60.
x=\frac{-\left(-8\right)±2}{2\times 5}
Take the square root of 4.
x=\frac{8±2}{2\times 5}
The opposite of -8 is 8.
x=\frac{8±2}{10}
Multiply 2 times 5.
x=\frac{10}{10}
Now solve the equation x=\frac{8±2}{10} when ± is plus. Add 8 to 2.
x=1
Divide 10 by 10.
x=\frac{6}{10}
Now solve the equation x=\frac{8±2}{10} when ± is minus. Subtract 2 from 8.
x=\frac{3}{5}
Reduce the fraction \frac{6}{10} to lowest terms by extracting and canceling out 2.
x=1 x=\frac{3}{5}
The equation is now solved.
5x^{2}-8x=-3
Subtract 8x from both sides.
\frac{5x^{2}-8x}{5}=-\frac{3}{5}
Divide both sides by 5.
x^{2}-\frac{8}{5}x=-\frac{3}{5}
Dividing by 5 undoes the multiplication by 5.
x^{2}-\frac{8}{5}x+\left(-\frac{4}{5}\right)^{2}=-\frac{3}{5}+\left(-\frac{4}{5}\right)^{2}
Divide -\frac{8}{5}, the coefficient of the x term, by 2 to get -\frac{4}{5}. Then add the square of -\frac{4}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{8}{5}x+\frac{16}{25}=-\frac{3}{5}+\frac{16}{25}
Square -\frac{4}{5} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{8}{5}x+\frac{16}{25}=\frac{1}{25}
Add -\frac{3}{5} to \frac{16}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{4}{5}\right)^{2}=\frac{1}{25}
Factor x^{2}-\frac{8}{5}x+\frac{16}{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{4}{5}\right)^{2}}=\sqrt{\frac{1}{25}}
Take the square root of both sides of the equation.
x-\frac{4}{5}=\frac{1}{5} x-\frac{4}{5}=-\frac{1}{5}
Simplify.
x=1 x=\frac{3}{5}
Add \frac{4}{5} to both sides of the equation.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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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}