Solve 10x^2+12x+18

$$10 x ^ { 2 } + 12 x + 18$$

Factor

$$2\left(5x^{2}+6x+9\right)$$

Solution Steps

Factor out 2. Polynomial 5x^{2}+6x+9 is not factored since it does not have any rational roots.

$$2\left(5x^{2}+6x+9\right)$$

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.

$$10x^{2}+12x+18=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\times 10\times 18}}{2\times 10}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 10\times 18}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-12±\sqrt{144-40\times 18}}{2\times 10}$$

Multiply -40 times 18.

$$x=\frac{-12±\sqrt{144-720}}{2\times 10}$$

Add 144 to -720.

$$x=\frac{-12±\sqrt{-576}}{2\times 10}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Quadratic polynomial cannot be factored.

$$10x^{2}+12x+18$$

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 10

$$x ^ 2 +\frac{6}{5}x +\frac{9}{5} = 0$$

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

$$r + s = -\frac{6}{5} rs = \frac{9}{5}$$

Two numbers r and s sum up to -\frac{6}{5} exactly when the average of the two numbers is \frac{1}{2}*-\frac{6}{5} = -\frac{3}{5}. 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>

$$r = -\frac{3}{5} - u s = -\frac{3}{5} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{9}{5}

$$(-\frac{3}{5} - u) (-\frac{3}{5} + u) = \frac{9}{5}$$

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

$$\frac{9}{25} - u^2 = \frac{9}{5}$$

Simplify the expression by subtracting \frac{9}{25} on both sides

$$-u^2 = \frac{9}{5}-\frac{9}{25} = \frac{36}{25}$$

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

$$u^2 = -\frac{36}{25} u = \pm\sqrt{-\frac{36}{25}} = \pm \frac{6}{5}i$$

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

$$r =-\frac{3}{5} - \frac{6}{5}i = -0.600 - 1.200i s = -\frac{3}{5} + \frac{6}{5}i = -0.600 + 1.200i$$

Evaluate

$$10x^{2}+12x+18$$