Factor
-\left(x+8\right)\left(x+10\right)
Evaluate
-\left(x+8\right)\left(x+10\right)
Graph
Share
Copied to clipboard
a+b=-18 ab=-\left(-80\right)=80
Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-80. To find a and b, set up a system to be solved.
-1,-80 -2,-40 -4,-20 -5,-16 -8,-10
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 80.
-1-80=-81 -2-40=-42 -4-20=-24 -5-16=-21 -8-10=-18
Calculate the sum for each pair.
a=-8 b=-10
The solution is the pair that gives sum -18.
\left(-x^{2}-8x\right)+\left(-10x-80\right)
Rewrite -x^{2}-18x-80 as \left(-x^{2}-8x\right)+\left(-10x-80\right).
x\left(-x-8\right)+10\left(-x-8\right)
Factor out x in the first and 10 in the second group.
\left(-x-8\right)\left(x+10\right)
Factor out common term -x-8 by using distributive property.
-x^{2}-18x-80=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.
x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-1\right)\left(-80\right)}}{2\left(-1\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.
x=\frac{-\left(-18\right)±\sqrt{324-4\left(-1\right)\left(-80\right)}}{2\left(-1\right)}
Square -18.
x=\frac{-\left(-18\right)±\sqrt{324+4\left(-80\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-\left(-18\right)±\sqrt{324-320}}{2\left(-1\right)}
Multiply 4 times -80.
x=\frac{-\left(-18\right)±\sqrt{4}}{2\left(-1\right)}
Add 324 to -320.
x=\frac{-\left(-18\right)±2}{2\left(-1\right)}
Take the square root of 4.
x=\frac{18±2}{2\left(-1\right)}
The opposite of -18 is 18.
x=\frac{18±2}{-2}
Multiply 2 times -1.
x=\frac{20}{-2}
Now solve the equation x=\frac{18±2}{-2} when ± is plus. Add 18 to 2.
x=-10
Divide 20 by -2.
x=\frac{16}{-2}
Now solve the equation x=\frac{18±2}{-2} when ± is minus. Subtract 2 from 18.
x=-8
Divide 16 by -2.
-x^{2}-18x-80=-\left(x-\left(-10\right)\right)\left(x-\left(-8\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -10 for x_{1} and -8 for x_{2}.
-x^{2}-18x-80=-\left(x+10\right)\left(x+8\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 +18x +80 = 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 = -18 rs = 80
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 = -9 - u s = -9 + u
Two numbers r and s sum up to -18 exactly when the average of the two numbers is \frac{1}{2}*-18 = -9. 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>
(-9 - u) (-9 + u) = 80
To solve for unknown quantity u, substitute these in the product equation rs = 80
81 - u^2 = 80
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 80-81 = -1
Simplify the expression by subtracting 81 on both sides
u^2 = 1 u = \pm\sqrt{1} = \pm 1
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-9 - 1 = -10 s = -9 + 1 = -8
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
Examples
Quadratic equation
{ 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}