Solve for a
a=2
a = -\frac{14}{3} = -4\frac{2}{3} \approx -4.666666667
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9a^{2}+24a+16=100
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3a+4\right)^{2}.
9a^{2}+24a+16-100=0
Subtract 100 from both sides.
9a^{2}+24a-84=0
Subtract 100 from 16 to get -84.
3a^{2}+8a-28=0
Divide both sides by 3.
a+b=8 ab=3\left(-28\right)=-84
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3a^{2}+aa+ba-28. To find a and b, set up a system to be solved.
-1,84 -2,42 -3,28 -4,21 -6,14 -7,12
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -84.
-1+84=83 -2+42=40 -3+28=25 -4+21=17 -6+14=8 -7+12=5
Calculate the sum for each pair.
a=-6 b=14
The solution is the pair that gives sum 8.
\left(3a^{2}-6a\right)+\left(14a-28\right)
Rewrite 3a^{2}+8a-28 as \left(3a^{2}-6a\right)+\left(14a-28\right).
3a\left(a-2\right)+14\left(a-2\right)
Factor out 3a in the first and 14 in the second group.
\left(a-2\right)\left(3a+14\right)
Factor out common term a-2 by using distributive property.
a=2 a=-\frac{14}{3}
To find equation solutions, solve a-2=0 and 3a+14=0.
9a^{2}+24a+16=100
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3a+4\right)^{2}.
9a^{2}+24a+16-100=0
Subtract 100 from both sides.
9a^{2}+24a-84=0
Subtract 100 from 16 to get -84.
a=\frac{-24±\sqrt{24^{2}-4\times 9\left(-84\right)}}{2\times 9}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 24 for b, and -84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
a=\frac{-24±\sqrt{576-4\times 9\left(-84\right)}}{2\times 9}
Square 24.
a=\frac{-24±\sqrt{576-36\left(-84\right)}}{2\times 9}
Multiply -4 times 9.
a=\frac{-24±\sqrt{576+3024}}{2\times 9}
Multiply -36 times -84.
a=\frac{-24±\sqrt{3600}}{2\times 9}
Add 576 to 3024.
a=\frac{-24±60}{2\times 9}
Take the square root of 3600.
a=\frac{-24±60}{18}
Multiply 2 times 9.
a=\frac{36}{18}
Now solve the equation a=\frac{-24±60}{18} when ± is plus. Add -24 to 60.
a=2
Divide 36 by 18.
a=-\frac{84}{18}
Now solve the equation a=\frac{-24±60}{18} when ± is minus. Subtract 60 from -24.
a=-\frac{14}{3}
Reduce the fraction \frac{-84}{18} to lowest terms by extracting and canceling out 6.
a=2 a=-\frac{14}{3}
The equation is now solved.
9a^{2}+24a+16=100
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3a+4\right)^{2}.
9a^{2}+24a=100-16
Subtract 16 from both sides.
9a^{2}+24a=84
Subtract 16 from 100 to get 84.
\frac{9a^{2}+24a}{9}=\frac{84}{9}
Divide both sides by 9.
a^{2}+\frac{24}{9}a=\frac{84}{9}
Dividing by 9 undoes the multiplication by 9.
a^{2}+\frac{8}{3}a=\frac{84}{9}
Reduce the fraction \frac{24}{9} to lowest terms by extracting and canceling out 3.
a^{2}+\frac{8}{3}a=\frac{28}{3}
Reduce the fraction \frac{84}{9} to lowest terms by extracting and canceling out 3.
a^{2}+\frac{8}{3}a+\left(\frac{4}{3}\right)^{2}=\frac{28}{3}+\left(\frac{4}{3}\right)^{2}
Divide \frac{8}{3}, the coefficient of the x term, by 2 to get \frac{4}{3}. Then add the square of \frac{4}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
a^{2}+\frac{8}{3}a+\frac{16}{9}=\frac{28}{3}+\frac{16}{9}
Square \frac{4}{3} by squaring both the numerator and the denominator of the fraction.
a^{2}+\frac{8}{3}a+\frac{16}{9}=\frac{100}{9}
Add \frac{28}{3} to \frac{16}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(a+\frac{4}{3}\right)^{2}=\frac{100}{9}
Factor a^{2}+\frac{8}{3}a+\frac{16}{9}. 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{4}{3}\right)^{2}}=\sqrt{\frac{100}{9}}
Take the square root of both sides of the equation.
a+\frac{4}{3}=\frac{10}{3} a+\frac{4}{3}=-\frac{10}{3}
Simplify.
a=2 a=-\frac{14}{3}
Subtract \frac{4}{3} from 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
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