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x^{2}+2x-35=0
Divide both sides by 2.
a+b=2 ab=1\left(-35\right)=-35
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-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 positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -35.
-1+35=34 -5+7=2
Calculate the sum for each pair.
a=-5 b=7
The solution is the pair that gives sum 2.
\left(x^{2}-5x\right)+\left(7x-35\right)
Rewrite x^{2}+2x-35 as \left(x^{2}-5x\right)+\left(7x-35\right).
x\left(x-5\right)+7\left(x-5\right)
Factor out x in the first and 7 in the second group.
\left(x-5\right)\left(x+7\right)
Factor out common term x-5 by using distributive property.
x=5 x=-7
To find equation solutions, solve x-5=0 and x+7=0.
2x^{2}+4x-70=0
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{-4±\sqrt{4^{2}-4\times 2\left(-70\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 4 for b, and -70 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-4±\sqrt{16-4\times 2\left(-70\right)}}{2\times 2}
Square 4.
x=\frac{-4±\sqrt{16-8\left(-70\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-4±\sqrt{16+560}}{2\times 2}
Multiply -8 times -70.
x=\frac{-4±\sqrt{576}}{2\times 2}
Add 16 to 560.
x=\frac{-4±24}{2\times 2}
Take the square root of 576.
x=\frac{-4±24}{4}
Multiply 2 times 2.
x=\frac{20}{4}
Now solve the equation x=\frac{-4±24}{4} when ± is plus. Add -4 to 24.
x=5
Divide 20 by 4.
x=-\frac{28}{4}
Now solve the equation x=\frac{-4±24}{4} when ± is minus. Subtract 24 from -4.
x=-7
Divide -28 by 4.
x=5 x=-7
The equation is now solved.
2x^{2}+4x-70=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
2x^{2}+4x-70-\left(-70\right)=-\left(-70\right)
Add 70 to both sides of the equation.
2x^{2}+4x=-\left(-70\right)
Subtracting -70 from itself leaves 0.
2x^{2}+4x=70
Subtract -70 from 0.
\frac{2x^{2}+4x}{2}=\frac{70}{2}
Divide both sides by 2.
x^{2}+\frac{4}{2}x=\frac{70}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}+2x=\frac{70}{2}
Divide 4 by 2.
x^{2}+2x=35
Divide 70 by 2.
x^{2}+2x+1^{2}=35+1^{2}
Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+2x+1=35+1
Square 1.
x^{2}+2x+1=36
Add 35 to 1.
\left(x+1\right)^{2}=36
Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x+1\right)^{2}}=\sqrt{36}
Take the square root of both sides of the equation.
x+1=6 x+1=-6
Simplify.
x=5 x=-7
Subtract 1 from both sides of the equation.
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.This is achieved by dividing both sides of the equation by 2
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 = -7 s = -1 + 6 = 5
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.