Solve 18x^2+31x-40

$$18 x ^ { 2 } + 31 x - 40$$

Factor

$$18\left(x-\frac{-\sqrt{3841}-31}{36}\right)\left(x-\frac{\sqrt{3841}-31}{36}\right)$$

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.

$$18x^{2}+31x-40=0$$

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.

$$x=\frac{-31±\sqrt{31^{2}-4\times 18\left(-40\right)}}{2\times 18}$$

Square 31.

$$x=\frac{-31±\sqrt{961-4\times 18\left(-40\right)}}{2\times 18}$$

Multiply -4 times 18.

$$x=\frac{-31±\sqrt{961-72\left(-40\right)}}{2\times 18}$$

Multiply -72 times -40.

$$x=\frac{-31±\sqrt{961+2880}}{2\times 18}$$

Add 961 to 2880.

$$x=\frac{-31±\sqrt{3841}}{2\times 18}$$

Multiply 2 times 18.

$$x=\frac{-31±\sqrt{3841}}{36}$$

Now solve the equation x=\frac{-31±\sqrt{3841}}{36} when ± is plus. Add -31 to \sqrt{3841}.

$$x=\frac{\sqrt{3841}-31}{36}$$

Now solve the equation x=\frac{-31±\sqrt{3841}}{36} when ± is minus. Subtract \sqrt{3841} from -31.

$$x=\frac{-\sqrt{3841}-31}{36}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-31+\sqrt{3841}}{36} for x_{1} and \frac{-31-\sqrt{3841}}{36} for x_{2}.

$$18x^{2}+31x-40=18\left(x-\frac{\sqrt{3841}-31}{36}\right)\left(x-\frac{-\sqrt{3841}-31}{36}\right)$$

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.This is achieved by dividing both sides of the equation by 18

$$x ^ 2 +\frac{31}{18}x -\frac{20}{9} = 0$$

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

$$r + s = -\frac{31}{18} rs = -\frac{20}{9}$$

Two numbers r and s sum up to -\frac{31}{18} exactly when the average of the two numbers is \frac{1}{2}*-\frac{31}{18} = -\frac{31}{36}. 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>

$$r = -\frac{31}{36} - u s = -\frac{31}{36} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{20}{9}

$$(-\frac{31}{36} - u) (-\frac{31}{36} + u) = -\frac{20}{9}$$

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

$$\frac{961}{1296} - u^2 = -\frac{20}{9}$$

Simplify the expression by subtracting \frac{961}{1296} on both sides

$$-u^2 = -\frac{20}{9}-\frac{961}{1296} = -\frac{3841}{1296}$$

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

$$u^2 = \frac{3841}{1296} u = \pm\sqrt{\frac{3841}{1296}} = \pm \frac{\sqrt{3841}}{36}$$

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

$$r =-\frac{31}{36} - \frac{\sqrt{3841}}{36} = -2.583 s = -\frac{31}{36} + \frac{\sqrt{3841}}{36} = 0.860$$

Evaluate

$$18x^{2}+31x-40$$