Solve for a
a=\frac{1}{2}=0.5
a=4
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2a^{2}-9a=-4
Subtract 9a from both sides.
2a^{2}-9a+4=0
Add 4 to both sides.
a+b=-9 ab=2\times 4=8
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2a^{2}+aa+ba+4. To find a and b, set up a system to be solved.
-1,-8 -2,-4
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 8.
-1-8=-9 -2-4=-6
Calculate the sum for each pair.
a=-8 b=-1
The solution is the pair that gives sum -9.
\left(2a^{2}-8a\right)+\left(-a+4\right)
Rewrite 2a^{2}-9a+4 as \left(2a^{2}-8a\right)+\left(-a+4\right).
2a\left(a-4\right)-\left(a-4\right)
Factor out 2a in the first and -1 in the second group.
\left(a-4\right)\left(2a-1\right)
Factor out common term a-4 by using distributive property.
a=4 a=\frac{1}{2}
To find equation solutions, solve a-4=0 and 2a-1=0.
2a^{2}-9a=-4
Subtract 9a from both sides.
2a^{2}-9a+4=0
Add 4 to both sides.
a=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 2\times 4}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -9 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
a=\frac{-\left(-9\right)±\sqrt{81-4\times 2\times 4}}{2\times 2}
Square -9.
a=\frac{-\left(-9\right)±\sqrt{81-8\times 4}}{2\times 2}
Multiply -4 times 2.
a=\frac{-\left(-9\right)±\sqrt{81-32}}{2\times 2}
Multiply -8 times 4.
a=\frac{-\left(-9\right)±\sqrt{49}}{2\times 2}
Add 81 to -32.
a=\frac{-\left(-9\right)±7}{2\times 2}
Take the square root of 49.
a=\frac{9±7}{2\times 2}
The opposite of -9 is 9.
a=\frac{9±7}{4}
Multiply 2 times 2.
a=\frac{16}{4}
Now solve the equation a=\frac{9±7}{4} when ± is plus. Add 9 to 7.
a=4
Divide 16 by 4.
a=\frac{2}{4}
Now solve the equation a=\frac{9±7}{4} when ± is minus. Subtract 7 from 9.
a=\frac{1}{2}
Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.
a=4 a=\frac{1}{2}
The equation is now solved.
2a^{2}-9a=-4
Subtract 9a from both sides.
\frac{2a^{2}-9a}{2}=-\frac{4}{2}
Divide both sides by 2.
a^{2}-\frac{9}{2}a=-\frac{4}{2}
Dividing by 2 undoes the multiplication by 2.
a^{2}-\frac{9}{2}a=-2
Divide -4 by 2.
a^{2}-\frac{9}{2}a+\left(-\frac{9}{4}\right)^{2}=-2+\left(-\frac{9}{4}\right)^{2}
Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
a^{2}-\frac{9}{2}a+\frac{81}{16}=-2+\frac{81}{16}
Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.
a^{2}-\frac{9}{2}a+\frac{81}{16}=\frac{49}{16}
Add -2 to \frac{81}{16}.
\left(a-\frac{9}{4}\right)^{2}=\frac{49}{16}
Factor a^{2}-\frac{9}{2}a+\frac{81}{16}. 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(a-\frac{9}{4}\right)^{2}}=\sqrt{\frac{49}{16}}
Take the square root of both sides of the equation.
a-\frac{9}{4}=\frac{7}{4} a-\frac{9}{4}=-\frac{7}{4}
Simplify.
a=4 a=\frac{1}{2}
Add \frac{9}{4} to both sides of the equation.
Examples
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{ 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}