Factor
\left(a+3\right)\left(3a+1\right)
Evaluate
\left(a+3\right)\left(3a+1\right)
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p+q=10 pq=3\times 3=9
Factor the expression by grouping. First, the expression needs to be rewritten as 3a^{2}+pa+qa+3. To find p and q, set up a system to be solved.
1,9 3,3
Since pq is positive, p and q have the same sign. Since p+q is positive, p and q are both positive. List all such integer pairs that give product 9.
1+9=10 3+3=6
Calculate the sum for each pair.
p=1 q=9
The solution is the pair that gives sum 10.
\left(3a^{2}+a\right)+\left(9a+3\right)
Rewrite 3a^{2}+10a+3 as \left(3a^{2}+a\right)+\left(9a+3\right).
a\left(3a+1\right)+3\left(3a+1\right)
Factor out a in the first and 3 in the second group.
\left(3a+1\right)\left(a+3\right)
Factor out common term 3a+1 by using distributive property.
3a^{2}+10a+3=0
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
a=\frac{-10±\sqrt{10^{2}-4\times 3\times 3}}{2\times 3}
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.
a=\frac{-10±\sqrt{100-4\times 3\times 3}}{2\times 3}
Square 10.
a=\frac{-10±\sqrt{100-12\times 3}}{2\times 3}
Multiply -4 times 3.
a=\frac{-10±\sqrt{100-36}}{2\times 3}
Multiply -12 times 3.
a=\frac{-10±\sqrt{64}}{2\times 3}
Add 100 to -36.
a=\frac{-10±8}{2\times 3}
Take the square root of 64.
a=\frac{-10±8}{6}
Multiply 2 times 3.
a=-\frac{2}{6}
Now solve the equation a=\frac{-10±8}{6} when ± is plus. Add -10 to 8.
a=-\frac{1}{3}
Reduce the fraction \frac{-2}{6} to lowest terms by extracting and canceling out 2.
a=-\frac{18}{6}
Now solve the equation a=\frac{-10±8}{6} when ± is minus. Subtract 8 from -10.
a=-3
Divide -18 by 6.
3a^{2}+10a+3=3\left(a-\left(-\frac{1}{3}\right)\right)\left(a-\left(-3\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{3} for x_{1} and -3 for x_{2}.
3a^{2}+10a+3=3\left(a+\frac{1}{3}\right)\left(a+3\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
3a^{2}+10a+3=3\times \frac{3a+1}{3}\left(a+3\right)
Add \frac{1}{3} to a by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
3a^{2}+10a+3=\left(3a+1\right)\left(a+3\right)
Cancel out 3, the greatest common factor in 3 and 3.
x ^ 2 +\frac{10}{3}x +1 = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 3
r + s = -\frac{10}{3} rs = 1
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = -\frac{5}{3} - u s = -\frac{5}{3} + u
Two numbers r and s sum up to -\frac{10}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{10}{3} = -\frac{5}{3}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(-\frac{5}{3} - u) (-\frac{5}{3} + u) = 1
To solve for unknown quantity u, substitute these in the product equation rs = 1
\frac{25}{9} - u^2 = 1
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 1-\frac{25}{9} = -\frac{16}{9}
Simplify the expression by subtracting \frac{25}{9} on both sides
u^2 = \frac{16}{9} u = \pm\sqrt{\frac{16}{9}} = \pm \frac{4}{3}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{5}{3} - \frac{4}{3} = -3 s = -\frac{5}{3} + \frac{4}{3} = -0.333
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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y = 3x + 4
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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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