Solve x^2-37x-650

$$x ^ { 2 } - 37 x - 650$$

Factor

$$\left(x-50\right)\left(x+13\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-650. To find a and b, set up a system to be solved.

$$a+b=-37 ab=1\left(-650\right)=-650$$

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 -650.

$$1,-650 2,-325 5,-130 10,-65 13,-50 25,-26$$

Calculate the sum for each pair.

$$1-650=-649 2-325=-323 5-130=-125 10-65=-55 13-50=-37 25-26=-1$$

The solution is the pair that gives sum -37.

$$a=-50 b=13$$

Rewrite x^{2}-37x-650 as \left(x^{2}-50x\right)+\left(13x-650\right).

$$\left(x^{2}-50x\right)+\left(13x-650\right)$$

Factor out x in the first and 13 in the second group.

$$x\left(x-50\right)+13\left(x-50\right)$$

Factor out common term x-50 by using distributive property.

$$\left(x-50\right)\left(x+13\right)$$

Steps Using the Quadratic Formula

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.

$$x^{2}-37x-650=0$$

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.

$$x=\frac{-\left(-37\right)±\sqrt{\left(-37\right)^{2}-4\left(-650\right)}}{2}$$

Square -37.

$$x=\frac{-\left(-37\right)±\sqrt{1369-4\left(-650\right)}}{2}$$

Multiply -4 times -650.

$$x=\frac{-\left(-37\right)±\sqrt{1369+2600}}{2}$$

Add 1369 to 2600.

$$x=\frac{-\left(-37\right)±\sqrt{3969}}{2}$$

Take the square root of 3969.

$$x=\frac{-\left(-37\right)±63}{2}$$

The opposite of -37 is 37.

$$x=\frac{37±63}{2}$$

Now solve the equation x=\frac{37±63}{2} when ± is plus. Add 37 to 63.

$$x=\frac{100}{2}$$

Divide 100 by 2.

$$x=50$$

Now solve the equation x=\frac{37±63}{2} when ± is minus. Subtract 63 from 37.

$$x=-\frac{26}{2}$$

Divide -26 by 2.

$$x=-13$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 50 for x_{1} and -13 for x_{2}.

$$x^{2}-37x-650=\left(x-50\right)\left(x-\left(-13\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$x^{2}-37x-650=\left(x-50\right)\left(x+13\right)$$

Steps Using Direct Factoring Method

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.

$$x ^ 2 -37x -650 = 0$$

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

$$r + s = 37 rs = -650$$

Two numbers r and s sum up to 37 exactly when the average of the two numbers is \frac{1}{2}*37 = \frac{37}{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>

$$r = \frac{37}{2} - u s = \frac{37}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -650

$$(\frac{37}{2} - u) (\frac{37}{2} + u) = -650$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{1369}{4} - u^2 = -650$$

Simplify the expression by subtracting \frac{1369}{4} on both sides

$$-u^2 = -650-\frac{1369}{4} = -\frac{3969}{4}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{3969}{4} u = \pm\sqrt{\frac{3969}{4}} = \pm \frac{63}{2}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{37}{2} - \frac{63}{2} = -13 s = \frac{37}{2} + \frac{63}{2} = 50$$

Evaluate

$$\left(x-50\right)\left(x+13\right)$$