Solve 6x^2+24x+18

$$6 x ^ { 2 } + 24 x + 18$$

Factor

$$6\left(x+1\right)\left(x+3\right)$$

Solution Steps

Factor out 6.

$$6\left(x^{2}+4x+3\right)$$

Consider x^{2}+4x+3. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

$$a+b=4 ab=1\times 3=3$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=3$$

Rewrite x^{2}+4x+3 as \left(x^{2}+x\right)+\left(3x+3\right).

$$\left(x^{2}+x\right)+\left(3x+3\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(x+1\right)+3\left(x+1\right)$$

Factor out common term x+1 by using distributive property.

$$\left(x+1\right)\left(x+3\right)$$

Rewrite the complete factored expression.

$$6\left(x+1\right)\left(x+3\right)$$

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.

$$6x^{2}+24x+18=0$$

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.

$$x=\frac{-24±\sqrt{24^{2}-4\times 6\times 18}}{2\times 6}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 6\times 18}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-24±\sqrt{576-24\times 18}}{2\times 6}$$

Multiply -24 times 18.

$$x=\frac{-24±\sqrt{576-432}}{2\times 6}$$

Add 576 to -432.

$$x=\frac{-24±\sqrt{144}}{2\times 6}$$

Take the square root of 144.

$$x=\frac{-24±12}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-24±12}{12}$$

Now solve the equation x=\frac{-24±12}{12} when ± is plus. Add -24 to 12.

$$x=-\frac{12}{12}$$

Divide -12 by 12.

$$x=-1$$

Now solve the equation x=\frac{-24±12}{12} when ± is minus. Subtract 12 from -24.

$$x=-\frac{36}{12}$$

Divide -36 by 12.

$$x=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and -3 for x_{2}.

$$6x^{2}+24x+18=6\left(x-\left(-1\right)\right)\left(x-\left(-3\right)\right)$$

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

$$6x^{2}+24x+18=6\left(x+1\right)\left(x+3\right)$$

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.This is achieved by dividing both sides of the equation by 6

$$x ^ 2 +4x +3 = 0$$

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

$$r + s = -4 rs = 3$$

Two numbers r and s sum up to -4 exactly when the average of the two numbers is \frac{1}{2}*-4 = -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>

$$r = -2 - u s = -2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 3

$$(-2 - u) (-2 + u) = 3$$

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

$$4 - u^2 = 3$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = 3-4 = -1$$

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

$$u^2 = 1 u = \pm\sqrt{1} = \pm 1$$

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

$$r =-2 - 1 = -3 s = -2 + 1 = -1$$

Evaluate

$$6\left(x+1\right)\left(x+3\right)$$