Factor
-3\left(q-6\right)\left(q+5\right)
Evaluate
-3\left(q-6\right)\left(q+5\right)
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3\left(-q^{2}+q+30\right)
Factor out 3.
a+b=1 ab=-30=-30
Consider -q^{2}+q+30. Factor the expression by grouping. First, the expression needs to be rewritten as -q^{2}+aq+bq+30. To find a and b, set up a system to be solved.
-1,30 -2,15 -3,10 -5,6
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 -30.
-1+30=29 -2+15=13 -3+10=7 -5+6=1
Calculate the sum for each pair.
a=6 b=-5
The solution is the pair that gives sum 1.
\left(-q^{2}+6q\right)+\left(-5q+30\right)
Rewrite -q^{2}+q+30 as \left(-q^{2}+6q\right)+\left(-5q+30\right).
-q\left(q-6\right)-5\left(q-6\right)
Factor out -q in the first and -5 in the second group.
\left(q-6\right)\left(-q-5\right)
Factor out common term q-6 by using distributive property.
3\left(q-6\right)\left(-q-5\right)
Rewrite the complete factored expression.
-3q^{2}+3q+90=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.
q=\frac{-3±\sqrt{3^{2}-4\left(-3\right)\times 90}}{2\left(-3\right)}
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.
q=\frac{-3±\sqrt{9-4\left(-3\right)\times 90}}{2\left(-3\right)}
Square 3.
q=\frac{-3±\sqrt{9+12\times 90}}{2\left(-3\right)}
Multiply -4 times -3.
q=\frac{-3±\sqrt{9+1080}}{2\left(-3\right)}
Multiply 12 times 90.
q=\frac{-3±\sqrt{1089}}{2\left(-3\right)}
Add 9 to 1080.
q=\frac{-3±33}{2\left(-3\right)}
Take the square root of 1089.
q=\frac{-3±33}{-6}
Multiply 2 times -3.
q=\frac{30}{-6}
Now solve the equation q=\frac{-3±33}{-6} when ± is plus. Add -3 to 33.
q=-5
Divide 30 by -6.
q=-\frac{36}{-6}
Now solve the equation q=\frac{-3±33}{-6} when ± is minus. Subtract 33 from -3.
q=6
Divide -36 by -6.
-3q^{2}+3q+90=-3\left(q-\left(-5\right)\right)\left(q-6\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -5 for x_{1} and 6 for x_{2}.
-3q^{2}+3q+90=-3\left(q+5\right)\left(q-6\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 -1x -30 = 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 = 1 rs = -30
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{1}{2} - u s = \frac{1}{2} + u
Two numbers r and s sum up to 1 exactly when the average of the two numbers is \frac{1}{2}*1 = \frac{1}{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{1}{2} - u) (\frac{1}{2} + u) = -30
To solve for unknown quantity u, substitute these in the product equation rs = -30
\frac{1}{4} - u^2 = -30
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -30-\frac{1}{4} = -\frac{121}{4}
Simplify the expression by subtracting \frac{1}{4} on both sides
u^2 = \frac{121}{4} u = \pm\sqrt{\frac{121}{4}} = \pm \frac{11}{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{1}{2} - \frac{11}{2} = -5 s = \frac{1}{2} + \frac{11}{2} = 6
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
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}