Factor
\left(A-9\right)\left(A+7\right)
Evaluate
\left(A-9\right)\left(A+7\right)
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a+b=-2 ab=1\left(-63\right)=-63
Factor the expression by grouping. First, the expression needs to be rewritten as A^{2}+aA+bA-63. To find a and b, set up a system to be solved.
1,-63 3,-21 7,-9
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 -63.
1-63=-62 3-21=-18 7-9=-2
Calculate the sum for each pair.
a=-9 b=7
The solution is the pair that gives sum -2.
\left(A^{2}-9A\right)+\left(7A-63\right)
Rewrite A^{2}-2A-63 as \left(A^{2}-9A\right)+\left(7A-63\right).
A\left(A-9\right)+7\left(A-9\right)
Factor out A in the first and 7 in the second group.
\left(A-9\right)\left(A+7\right)
Factor out common term A-9 by using distributive property.
A^{2}-2A-63=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(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-63\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(-2\right)±\sqrt{4-4\left(-63\right)}}{2}
Square -2.
A=\frac{-\left(-2\right)±\sqrt{4+252}}{2}
Multiply -4 times -63.
A=\frac{-\left(-2\right)±\sqrt{256}}{2}
Add 4 to 252.
A=\frac{-\left(-2\right)±16}{2}
Take the square root of 256.
A=\frac{2±16}{2}
The opposite of -2 is 2.
A=\frac{18}{2}
Now solve the equation A=\frac{2±16}{2} when ± is plus. Add 2 to 16.
A=9
Divide 18 by 2.
A=-\frac{14}{2}
Now solve the equation A=\frac{2±16}{2} when ± is minus. Subtract 16 from 2.
A=-7
Divide -14 by 2.
A^{2}-2A-63=\left(A-9\right)\left(A-\left(-7\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 -7 for x_{2}.
A^{2}-2A-63=\left(A-9\right)\left(A+7\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 -2x -63 = 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 = 2 rs = -63
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 = 1 - u s = 1 + u
Two numbers r and s sum up to 2 exactly when the average of the two numbers is \frac{1}{2}*2 = 1. 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>
(1 - u) (1 + u) = -63
To solve for unknown quantity u, substitute these in the product equation rs = -63
1 - u^2 = -63
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -63-1 = -64
Simplify the expression by subtracting 1 on both sides
u^2 = 64 u = \pm\sqrt{64} = \pm 8
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =1 - 8 = -7 s = 1 + 8 = 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}