Solve x^2+54x+648

$$x ^ { 2 } + 54 x + 648$$

Factor

$$\left(x+18\right)\left(x+36\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+648. To find a and b, set up a system to be solved.

$$a+b=54 ab=1\times 648=648$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 648.

$$1,648 2,324 3,216 4,162 6,108 8,81 9,72 12,54 18,36 24,27$$

Calculate the sum for each pair.

$$1+648=649 2+324=326 3+216=219 4+162=166 6+108=114 8+81=89 9+72=81 12+54=66 18+36=54 24+27=51$$

The solution is the pair that gives sum 54.

$$a=18 b=36$$

Rewrite x^{2}+54x+648 as \left(x^{2}+18x\right)+\left(36x+648\right).

$$\left(x^{2}+18x\right)+\left(36x+648\right)$$

Factor out x in the first and 36 in the second group.

$$x\left(x+18\right)+36\left(x+18\right)$$

Factor out common term x+18 by using distributive property.

$$\left(x+18\right)\left(x+36\right)$$

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.

$$x^{2}+54x+648=0$$

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.

$$x=\frac{-54±\sqrt{54^{2}-4\times 648}}{2}$$

Square 54.

$$x=\frac{-54±\sqrt{2916-4\times 648}}{2}$$

Multiply -4 times 648.

$$x=\frac{-54±\sqrt{2916-2592}}{2}$$

Add 2916 to -2592.

$$x=\frac{-54±\sqrt{324}}{2}$$

Take the square root of 324.

$$x=\frac{-54±18}{2}$$

Now solve the equation x=\frac{-54±18}{2} when ± is plus. Add -54 to 18.

$$x=-\frac{36}{2}$$

Divide -36 by 2.

$$x=-18$$

Now solve the equation x=\frac{-54±18}{2} when ± is minus. Subtract 18 from -54.

$$x=-\frac{72}{2}$$

Divide -72 by 2.

$$x=-36$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -18 for x_{1} and -36 for x_{2}.

$$x^{2}+54x+648=\left(x-\left(-18\right)\right)\left(x-\left(-36\right)\right)$$

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

$$x^{2}+54x+648=\left(x+18\right)\left(x+36\right)$$

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.

$$x ^ 2 +54x +648 = 0$$

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

$$r + s = -54 rs = 648$$

Two numbers r and s sum up to -54 exactly when the average of the two numbers is \frac{1}{2}*-54 = -27. 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>

$$r = -27 - u s = -27 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 648

$$(-27 - u) (-27 + u) = 648$$

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

$$729 - u^2 = 648$$

Simplify the expression by subtracting 729 on both sides

$$-u^2 = 648-729 = -81$$

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

$$u^2 = 81 u = \pm\sqrt{81} = \pm 9$$

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

$$r =-27 - 9 = -36 s = -27 + 9 = -18$$

Evaluate

$$\left(x+18\right)\left(x+36\right)$$