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a+b=-2 ab=1\left(-35\right)=-35
Factor the expression by grouping. First, the expression needs to be rewritten as r^{2}+ar+br-35. To find a and b, set up a system to be solved.
1,-35 5,-7
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 -35.
1-35=-34 5-7=-2
Calculate the sum for each pair.
a=-7 b=5
The solution is the pair that gives sum -2.
\left(r^{2}-7r\right)+\left(5r-35\right)
Rewrite r^{2}-2r-35 as \left(r^{2}-7r\right)+\left(5r-35\right).
r\left(r-7\right)+5\left(r-7\right)
Factor out r in the first and 5 in the second group.
\left(r-7\right)\left(r+5\right)
Factor out common term r-7 by using distributive property.
r^{2}-2r-35=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.
r=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-35\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.
r=\frac{-\left(-2\right)±\sqrt{4-4\left(-35\right)}}{2}
Square -2.
r=\frac{-\left(-2\right)±\sqrt{4+140}}{2}
Multiply -4 times -35.
r=\frac{-\left(-2\right)±\sqrt{144}}{2}
Add 4 to 140.
r=\frac{-\left(-2\right)±12}{2}
Take the square root of 144.
r=\frac{2±12}{2}
The opposite of -2 is 2.
r=\frac{14}{2}
Now solve the equation r=\frac{2±12}{2} when ± is plus. Add 2 to 12.
r=7
Divide 14 by 2.
r=-\frac{10}{2}
Now solve the equation r=\frac{2±12}{2} when ± is minus. Subtract 12 from 2.
r=-5
Divide -10 by 2.
r^{2}-2r-35=\left(r-7\right)\left(r-\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 7 for x_{1} and -5 for x_{2}.
r^{2}-2r-35=\left(r-7\right)\left(r+5\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 -2x -35 = 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 = -35
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) = -35
To solve for unknown quantity u, substitute these in the product equation rs = -35
1 - u^2 = -35
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -35-1 = -36
Simplify the expression by subtracting 1 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 =1 - 6 = -5 s = 1 + 6 = 7
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.