Solve x^2-65x+900

$$x ^ { 2 } - 65 x + 900$$

Factor

$$\left(x-45\right)\left(x-20\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+900. To find a and b, set up a system to be solved.

$$a+b=-65 ab=1\times 900=900$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 900.

$$-1,-900 -2,-450 -3,-300 -4,-225 -5,-180 -6,-150 -9,-100 -10,-90 -12,-75 -15,-60 -18,-50 -20,-45 -25,-36 -30,-30$$

Calculate the sum for each pair.

$$-1-900=-901 -2-450=-452 -3-300=-303 -4-225=-229 -5-180=-185 -6-150=-156 -9-100=-109 -10-90=-100 -12-75=-87 -15-60=-75 -18-50=-68 -20-45=-65 -25-36=-61 -30-30=-60$$

The solution is the pair that gives sum -65.

$$a=-45 b=-20$$

Rewrite x^{2}-65x+900 as \left(x^{2}-45x\right)+\left(-20x+900\right).

$$\left(x^{2}-45x\right)+\left(-20x+900\right)$$

Factor out x in the first and -20 in the second group.

$$x\left(x-45\right)-20\left(x-45\right)$$

Factor out common term x-45 by using distributive property.

$$\left(x-45\right)\left(x-20\right)$$

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.

$$x^{2}-65x+900=0$$

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.

$$x=\frac{-\left(-65\right)±\sqrt{\left(-65\right)^{2}-4\times 900}}{2}$$

Square -65.

$$x=\frac{-\left(-65\right)±\sqrt{4225-4\times 900}}{2}$$

Multiply -4 times 900.

$$x=\frac{-\left(-65\right)±\sqrt{4225-3600}}{2}$$

Add 4225 to -3600.

$$x=\frac{-\left(-65\right)±\sqrt{625}}{2}$$

Take the square root of 625.

$$x=\frac{-\left(-65\right)±25}{2}$$

The opposite of -65 is 65.

$$x=\frac{65±25}{2}$$

Now solve the equation x=\frac{65±25}{2} when ± is plus. Add 65 to 25.

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

Divide 90 by 2.

$$x=45$$

Now solve the equation x=\frac{65±25}{2} when ± is minus. Subtract 25 from 65.

$$x=\frac{40}{2}$$

Divide 40 by 2.

$$x=20$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 45 for x_{1} and 20 for x_{2}.

$$x^{2}-65x+900=\left(x-45\right)\left(x-20\right)$$

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.

$$x ^ 2 -65x +900 = 0$$

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

$$r + s = 65 rs = 900$$

Two numbers r and s sum up to 65 exactly when the average of the two numbers is \frac{1}{2}*65 = \frac{65}{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>

$$r = \frac{65}{2} - u s = \frac{65}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 900

$$(\frac{65}{2} - u) (\frac{65}{2} + u) = 900$$

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

$$\frac{4225}{4} - u^2 = 900$$

Simplify the expression by subtracting \frac{4225}{4} on both sides

$$-u^2 = 900-\frac{4225}{4} = -\frac{625}{4}$$

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

$$u^2 = \frac{625}{4} u = \pm\sqrt{\frac{625}{4}} = \pm \frac{25}{2}$$

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

$$r =\frac{65}{2} - \frac{25}{2} = 20 s = \frac{65}{2} + \frac{25}{2} = 45$$

Evaluate

$$\left(x-45\right)\left(x-20\right)$$