Solve x^2+5x-1800

$$x ^ { 2 } + 5 x - 1800$$

Factor

$$\left(x-40\right)\left(x+45\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-1800. To find a and b, set up a system to be solved.

$$a+b=5 ab=1\left(-1800\right)=-1800$$

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 -1800.

$$-1,1800 -2,900 -3,600 -4,450 -5,360 -6,300 -8,225 -9,200 -10,180 -12,150 -15,120 -18,100 -20,90 -24,75 -25,72 -30,60 -36,50 -40,45$$

Calculate the sum for each pair.

$$-1+1800=1799 -2+900=898 -3+600=597 -4+450=446 -5+360=355 -6+300=294 -8+225=217 -9+200=191 -10+180=170 -12+150=138 -15+120=105 -18+100=82 -20+90=70 -24+75=51 -25+72=47 -30+60=30 -36+50=14 -40+45=5$$

The solution is the pair that gives sum 5.

$$a=-40 b=45$$

Rewrite x^{2}+5x-1800 as \left(x^{2}-40x\right)+\left(45x-1800\right).

$$\left(x^{2}-40x\right)+\left(45x-1800\right)$$

Factor out x in the first and 45 in the second group.

$$x\left(x-40\right)+45\left(x-40\right)$$

Factor out common term x-40 by using distributive property.

$$\left(x-40\right)\left(x+45\right)$$

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.

$$x^{2}+5x-1800=0$$

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.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1800\right)}}{2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-1800\right)}}{2}$$

Multiply -4 times -1800.

$$x=\frac{-5±\sqrt{25+7200}}{2}$$

Add 25 to 7200.

$$x=\frac{-5±\sqrt{7225}}{2}$$

Take the square root of 7225.

$$x=\frac{-5±85}{2}$$

Now solve the equation x=\frac{-5±85}{2} when ± is plus. Add -5 to 85.

$$x=\frac{80}{2}$$

Divide 80 by 2.

$$x=40$$

Now solve the equation x=\frac{-5±85}{2} when ± is minus. Subtract 85 from -5.

$$x=-\frac{90}{2}$$

Divide -90 by 2.

$$x=-45$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 40 for x_{1} and -45 for x_{2}.

$$x^{2}+5x-1800=\left(x-40\right)\left(x-\left(-45\right)\right)$$

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

$$x^{2}+5x-1800=\left(x-40\right)\left(x+45\right)$$

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.

$$x ^ 2 +5x -1800 = 0$$

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

$$r + s = -5 rs = -1800$$

Two numbers r and s sum up to -5 exactly when the average of the two numbers is \frac{1}{2}*-5 = -\frac{5}{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>

$$r = -\frac{5}{2} - u s = -\frac{5}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -1800

$$(-\frac{5}{2} - u) (-\frac{5}{2} + u) = -1800$$

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

$$\frac{25}{4} - u^2 = -1800$$

Simplify the expression by subtracting \frac{25}{4} on both sides

$$-u^2 = -1800-\frac{25}{4} = -\frac{7225}{4}$$

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

$$u^2 = \frac{7225}{4} u = \pm\sqrt{\frac{7225}{4}} = \pm \frac{85}{2}$$

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

$$r =-\frac{5}{2} - \frac{85}{2} = -45 s = -\frac{5}{2} + \frac{85}{2} = 40$$

Evaluate

$$\left(x-40\right)\left(x+45\right)$$