a+b=-1 ab=-756
To solve the equation, factor x^{2}-x-756 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.
1,-756 2,-378 3,-252 4,-189 6,-126 7,-108 9,-84 12,-63 14,-54 18,-42 21,-36 27,-28
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 -756.
1-756=-755 2-378=-376 3-252=-249 4-189=-185 6-126=-120 7-108=-101 9-84=-75 12-63=-51 14-54=-40 18-42=-24 21-36=-15 27-28=-1
Calculate the sum for each pair.
a=-28 b=27
The solution is the pair that gives sum -1.
\left(x-28\right)\left(x+27\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=28 x=-27
To find equation solutions, solve x-28=0 and x+27=0.
a+b=-1 ab=1\left(-756\right)=-756
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-756. To find a and b, set up a system to be solved.
1,-756 2,-378 3,-252 4,-189 6,-126 7,-108 9,-84 12,-63 14,-54 18,-42 21,-36 27,-28
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 -756.
1-756=-755 2-378=-376 3-252=-249 4-189=-185 6-126=-120 7-108=-101 9-84=-75 12-63=-51 14-54=-40 18-42=-24 21-36=-15 27-28=-1
Calculate the sum for each pair.
a=-28 b=27
The solution is the pair that gives sum -1.
\left(x^{2}-28x\right)+\left(27x-756\right)
Rewrite x^{2}-x-756 as \left(x^{2}-28x\right)+\left(27x-756\right).
x\left(x-28\right)+27\left(x-28\right)
Factor out x in the first and 27 in the second group.
\left(x-28\right)\left(x+27\right)
Factor out common term x-28 by using distributive property.
x=28 x=-27
To find equation solutions, solve x-28=0 and x+27=0.
x^{2}-x-756=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{-\left(-1\right)±\sqrt{1-4\left(-756\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -756 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-1\right)±\sqrt{1+3024}}{2}
Multiply -4 times -756.
x=\frac{-\left(-1\right)±\sqrt{3025}}{2}
Add 1 to 3024.
x=\frac{-\left(-1\right)±55}{2}
Take the square root of 3025.
x=\frac{1±55}{2}
The opposite of -1 is 1.
x=\frac{56}{2}
Now solve the equation x=\frac{1±55}{2} when ± is plus. Add 1 to 55.
x=-\frac{54}{2}
Now solve the equation x=\frac{1±55}{2} when ± is minus. Subtract 55 from 1.
x=-27
Divide -54 by 2.
x=28 x=-27
The equation is now solved.
x^{2}-x-756=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.
x^{2}-x-756-\left(-756\right)=-\left(-756\right)
Add 756 to both sides of the equation.
x^{2}-x=-\left(-756\right)
Subtracting -756 from itself leaves 0.
x^{2}-x=756
Subtract -756 from 0.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=756+\left(-\frac{1}{2}\right)^{2}
Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-x+\frac{1}{4}=756+\frac{1}{4}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-x+\frac{1}{4}=\frac{3025}{4}
Add 756 to \frac{1}{4}.
\left(x-\frac{1}{2}\right)^{2}=\frac{3025}{4}
Factor x^{2}-x+\frac{1}{4}. 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-\frac{1}{2}\right)^{2}}=\sqrt{\frac{3025}{4}}
Take the square root of both sides of the equation.
x-\frac{1}{2}=\frac{55}{2} x-\frac{1}{2}=-\frac{55}{2}
Simplify.
x=28 x=-27
Add \frac{1}{2} to both sides of the equation.
x ^ 2 -1x -756 = 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 = 1 rs = -756
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 = \frac{1}{2} - u s = \frac{1}{2} + u
Two numbers r and s sum up to 1 exactly when the average of the two numbers is \frac{1}{2}*1 = \frac{1}{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>
(\frac{1}{2} - u) (\frac{1}{2} + u) = -756
To solve for unknown quantity u, substitute these in the product equation rs = -756
\frac{1}{4} - u^2 = -756
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -756-\frac{1}{4} = -\frac{3025}{4}
Simplify the expression by subtracting \frac{1}{4} on both sides
u^2 = \frac{3025}{4} u = \pm\sqrt{\frac{3025}{4}} = \pm \frac{55}{2}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{1}{2} - \frac{55}{2} = -27 s = \frac{1}{2} + \frac{55}{2} = 28
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
