Solve for l
l=-18
l=21
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a+b=-3 ab=-378
To solve the equation, factor l^{2}-3l-378 using formula l^{2}+\left(a+b\right)l+ab=\left(l+a\right)\left(l+b\right). To find a and b, set up a system to be solved.
1,-378 2,-189 3,-126 6,-63 7,-54 9,-42 14,-27 18,-21
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -378.
1-378=-377 2-189=-187 3-126=-123 6-63=-57 7-54=-47 9-42=-33 14-27=-13 18-21=-3
Calculate the sum for each pair.
a=-21 b=18
The solution is the pair that gives sum -3.
\left(l-21\right)\left(l+18\right)
Rewrite factored expression \left(l+a\right)\left(l+b\right) using the obtained values.
l=21 l=-18
To find equation solutions, solve l-21=0 and l+18=0.
a+b=-3 ab=1\left(-378\right)=-378
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as l^{2}+al+bl-378. To find a and b, set up a system to be solved.
1,-378 2,-189 3,-126 6,-63 7,-54 9,-42 14,-27 18,-21
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -378.
1-378=-377 2-189=-187 3-126=-123 6-63=-57 7-54=-47 9-42=-33 14-27=-13 18-21=-3
Calculate the sum for each pair.
a=-21 b=18
The solution is the pair that gives sum -3.
\left(l^{2}-21l\right)+\left(18l-378\right)
Rewrite l^{2}-3l-378 as \left(l^{2}-21l\right)+\left(18l-378\right).
l\left(l-21\right)+18\left(l-21\right)
Factor out l in the first and 18 in the second group.
\left(l-21\right)\left(l+18\right)
Factor out common term l-21 by using distributive property.
l=21 l=-18
To find equation solutions, solve l-21=0 and l+18=0.
l^{2}-3l-378=0
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.
l=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-378\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -378 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
l=\frac{-\left(-3\right)±\sqrt{9-4\left(-378\right)}}{2}
Square -3.
l=\frac{-\left(-3\right)±\sqrt{9+1512}}{2}
Multiply -4 times -378.
l=\frac{-\left(-3\right)±\sqrt{1521}}{2}
Add 9 to 1512.
l=\frac{-\left(-3\right)±39}{2}
Take the square root of 1521.
l=\frac{3±39}{2}
The opposite of -3 is 3.
l=\frac{42}{2}
Now solve the equation l=\frac{3±39}{2} when ± is plus. Add 3 to 39.
l=21
Divide 42 by 2.
l=-\frac{36}{2}
Now solve the equation l=\frac{3±39}{2} when ± is minus. Subtract 39 from 3.
l=-18
Divide -36 by 2.
l=21 l=-18
The equation is now solved.
l^{2}-3l-378=0
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.
l^{2}-3l-378-\left(-378\right)=-\left(-378\right)
Add 378 to both sides of the equation.
l^{2}-3l=-\left(-378\right)
Subtracting -378 from itself leaves 0.
l^{2}-3l=378
Subtract -378 from 0.
l^{2}-3l+\left(-\frac{3}{2}\right)^{2}=378+\left(-\frac{3}{2}\right)^{2}
Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
l^{2}-3l+\frac{9}{4}=378+\frac{9}{4}
Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.
l^{2}-3l+\frac{9}{4}=\frac{1521}{4}
Add 378 to \frac{9}{4}.
\left(l-\frac{3}{2}\right)^{2}=\frac{1521}{4}
Factor l^{2}-3l+\frac{9}{4}. 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(l-\frac{3}{2}\right)^{2}}=\sqrt{\frac{1521}{4}}
Take the square root of both sides of the equation.
l-\frac{3}{2}=\frac{39}{2} l-\frac{3}{2}=-\frac{39}{2}
Simplify.
l=21 l=-18
Add \frac{3}{2} to both sides of the equation.
x ^ 2 -3x -378 = 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.
r + s = 3 rs = -378
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{3}{2} - u s = \frac{3}{2} + u
Two numbers r and s sum up to 3 exactly when the average of the two numbers is \frac{1}{2}*3 = \frac{3}{2}. 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{3}{2} - u) (\frac{3}{2} + u) = -378
To solve for unknown quantity u, substitute these in the product equation rs = -378
\frac{9}{4} - u^2 = -378
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -378-\frac{9}{4} = -\frac{1521}{4}
Simplify the expression by subtracting \frac{9}{4} on both sides
u^2 = \frac{1521}{4} u = \pm\sqrt{\frac{1521}{4}} = \pm \frac{39}{2}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{3}{2} - \frac{39}{2} = -18 s = \frac{3}{2} + \frac{39}{2} = 21
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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