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a+b=-14 ab=1\left(-15\right)=-15
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-15. To find a and b, set up a system to be solved.
1,-15 3,-5
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 -15.
1-15=-14 3-5=-2
Calculate the sum for each pair.
a=-15 b=1
The solution is the pair that gives sum -14.
\left(x^{2}-15x\right)+\left(x-15\right)
Rewrite x^{2}-14x-15 as \left(x^{2}-15x\right)+\left(x-15\right).
x\left(x-15\right)+x-15
Factor out x in x^{2}-15x.
\left(x-15\right)\left(x+1\right)
Factor out common term x-15 by using distributive property.
x^{2}-14x-15=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{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-15\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{-\left(-14\right)±\sqrt{196-4\left(-15\right)}}{2}
Square -14.
x=\frac{-\left(-14\right)±\sqrt{196+60}}{2}
Multiply -4 times -15.
x=\frac{-\left(-14\right)±\sqrt{256}}{2}
Add 196 to 60.
x=\frac{-\left(-14\right)±16}{2}
Take the square root of 256.
x=\frac{14±16}{2}
The opposite of -14 is 14.
x=\frac{30}{2}
Now solve the equation x=\frac{14±16}{2} when ± is plus. Add 14 to 16.
x=15
Divide 30 by 2.
x=-\frac{2}{2}
Now solve the equation x=\frac{14±16}{2} when ± is minus. Subtract 16 from 14.
x=-1
Divide -2 by 2.
x^{2}-14x-15=\left(x-15\right)\left(x-\left(-1\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 15 for x_{1} and -1 for x_{2}.
x^{2}-14x-15=\left(x-15\right)\left(x+1\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 -14x -15 = 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 = 14 rs = -15
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 = 7 - u s = 7 + u
Two numbers r and s sum up to 14 exactly when the average of the two numbers is \frac{1}{2}*14 = 7. 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>
(7 - u) (7 + u) = -15
To solve for unknown quantity u, substitute these in the product equation rs = -15
49 - u^2 = -15
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -15-49 = -64
Simplify the expression by subtracting 49 on both sides
u^2 = 64 u = \pm\sqrt{64} = \pm 8
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =7 - 8 = -1 s = 7 + 8 = 15
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.