Factor
\left(a-9\right)\left(a+5\right)
Evaluate
\left(a-9\right)\left(a+5\right)
Share
Copied to clipboard
p+q=-4 pq=1\left(-45\right)=-45
Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-45. To find p and q, set up a system to be solved.
1,-45 3,-15 5,-9
Since pq is negative, p and q have the opposite signs. Since p+q is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -45.
1-45=-44 3-15=-12 5-9=-4
Calculate the sum for each pair.
p=-9 q=5
The solution is the pair that gives sum -4.
\left(a^{2}-9a\right)+\left(5a-45\right)
Rewrite a^{2}-4a-45 as \left(a^{2}-9a\right)+\left(5a-45\right).
a\left(a-9\right)+5\left(a-9\right)
Factor out a in the first and 5 in the second group.
\left(a-9\right)\left(a+5\right)
Factor out common term a-9 by using distributive property.
a^{2}-4a-45=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{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-45\right)}}{2}
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{-\left(-4\right)±\sqrt{16-4\left(-45\right)}}{2}
Square -4.
a=\frac{-\left(-4\right)±\sqrt{16+180}}{2}
Multiply -4 times -45.
a=\frac{-\left(-4\right)±\sqrt{196}}{2}
Add 16 to 180.
a=\frac{-\left(-4\right)±14}{2}
Take the square root of 196.
a=\frac{4±14}{2}
The opposite of -4 is 4.
a=\frac{18}{2}
Now solve the equation a=\frac{4±14}{2} when ± is plus. Add 4 to 14.
a=9
Divide 18 by 2.
a=-\frac{10}{2}
Now solve the equation a=\frac{4±14}{2} when ± is minus. Subtract 14 from 4.
a=-5
Divide -10 by 2.
a^{2}-4a-45=\left(a-9\right)\left(a-\left(-5\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 9 for x_{1} and -5 for x_{2}.
a^{2}-4a-45=\left(a-9\right)\left(a+5\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 -4x -45 = 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 = 4 rs = -45
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 = 2 - u s = 2 + u
Two numbers r and s sum up to 4 exactly when the average of the two numbers is \frac{1}{2}*4 = 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>
(2 - u) (2 + u) = -45
To solve for unknown quantity u, substitute these in the product equation rs = -45
4 - u^2 = -45
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -45-4 = -49
Simplify the expression by subtracting 4 on both sides
u^2 = 49 u = \pm\sqrt{49} = \pm 7
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =2 - 7 = -5 s = 2 + 7 = 9
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}