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p+q=4 pq=1\left(-32\right)=-32
Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-32. To find p and q, set up a system to be solved.
-1,32 -2,16 -4,8
Since pq is negative, p and q have the opposite signs. Since p+q is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -32.
-1+32=31 -2+16=14 -4+8=4
Calculate the sum for each pair.
p=-4 q=8
The solution is the pair that gives sum 4.
\left(a^{2}-4a\right)+\left(8a-32\right)
Rewrite a^{2}+4a-32 as \left(a^{2}-4a\right)+\left(8a-32\right).
a\left(a-4\right)+8\left(a-4\right)
Factor out a in the first and 8 in the second group.
\left(a-4\right)\left(a+8\right)
Factor out common term a-4 by using distributive property.
a^{2}+4a-32=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{-4±\sqrt{4^{2}-4\left(-32\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{-4±\sqrt{16-4\left(-32\right)}}{2}
Square 4.
a=\frac{-4±\sqrt{16+128}}{2}
Multiply -4 times -32.
a=\frac{-4±\sqrt{144}}{2}
Add 16 to 128.
a=\frac{-4±12}{2}
Take the square root of 144.
a=\frac{8}{2}
Now solve the equation a=\frac{-4±12}{2} when ± is plus. Add -4 to 12.
a=4
Divide 8 by 2.
a=-\frac{16}{2}
Now solve the equation a=\frac{-4±12}{2} when ± is minus. Subtract 12 from -4.
a=-8
Divide -16 by 2.
a^{2}+4a-32=\left(a-4\right)\left(a-\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 4 for x_{1} and -8 for x_{2}.
a^{2}+4a-32=\left(a-4\right)\left(a+8\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 +4x -32 = 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 = -32
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) = -32
To solve for unknown quantity u, substitute these in the product equation rs = -32
4 - u^2 = -32
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -32-4 = -36
Simplify the expression by subtracting 4 on both sides
u^2 = 36 u = \pm\sqrt{36} = \pm 6
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-2 - 6 = -8 s = -2 + 6 = 4
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.