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a+b=28 ab=1\left(-29\right)=-29
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-29. To find a and b, set up a system to be solved.
a=-1 b=29
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. The only such pair is the system solution.
\left(x^{2}-x\right)+\left(29x-29\right)
Rewrite x^{2}+28x-29 as \left(x^{2}-x\right)+\left(29x-29\right).
x\left(x-1\right)+29\left(x-1\right)
Factor out x in the first and 29 in the second group.
\left(x-1\right)\left(x+29\right)
Factor out common term x-1 by using distributive property.
x^{2}+28x-29=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.
x=\frac{-28±\sqrt{28^{2}-4\left(-29\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.
x=\frac{-28±\sqrt{784-4\left(-29\right)}}{2}
Square 28.
x=\frac{-28±\sqrt{784+116}}{2}
Multiply -4 times -29.
x=\frac{-28±\sqrt{900}}{2}
Add 784 to 116.
x=\frac{-28±30}{2}
Take the square root of 900.
x=\frac{2}{2}
Now solve the equation x=\frac{-28±30}{2} when ± is plus. Add -28 to 30.
x=1
Divide 2 by 2.
x=-\frac{58}{2}
Now solve the equation x=\frac{-28±30}{2} when ± is minus. Subtract 30 from -28.
x=-29
Divide -58 by 2.
x^{2}+28x-29=\left(x-1\right)\left(x-\left(-29\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and -29 for x_{2}.
x^{2}+28x-29=\left(x-1\right)\left(x+29\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 +28x -29 = 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 = -28 rs = -29
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 = -14 - u s = -14 + u
Two numbers r and s sum up to -28 exactly when the average of the two numbers is \frac{1}{2}*-28 = -14. 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>
(-14 - u) (-14 + u) = -29
To solve for unknown quantity u, substitute these in the product equation rs = -29
196 - u^2 = -29
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -29-196 = -225
Simplify the expression by subtracting 196 on both sides
u^2 = 225 u = \pm\sqrt{225} = \pm 15
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-14 - 15 = -29 s = -14 + 15 = 1
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.