Solve 142x^2-3x-9

$$142 x ^ { 2 } - 3 x - 9$$

Factor

$$142\left(x-\frac{3-3\sqrt{569}}{284}\right)\left(x-\frac{3\sqrt{569}+3}{284}\right)$$

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.

$$142x^{2}-3x-9=0$$

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.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 142\left(-9\right)}}{2\times 142}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 142\left(-9\right)}}{2\times 142}$$

Multiply -4 times 142.

$$x=\frac{-\left(-3\right)±\sqrt{9-568\left(-9\right)}}{2\times 142}$$

Multiply -568 times -9.

$$x=\frac{-\left(-3\right)±\sqrt{9+5112}}{2\times 142}$$

Add 9 to 5112.

$$x=\frac{-\left(-3\right)±\sqrt{5121}}{2\times 142}$$

Take the square root of 5121.

$$x=\frac{-\left(-3\right)±3\sqrt{569}}{2\times 142}$$

The opposite of -3 is 3.

$$x=\frac{3±3\sqrt{569}}{2\times 142}$$

Multiply 2 times 142.

$$x=\frac{3±3\sqrt{569}}{284}$$

Now solve the equation x=\frac{3±3\sqrt{569}}{284} when ± is plus. Add 3 to 3\sqrt{569}.

$$x=\frac{3\sqrt{569}+3}{284}$$

Now solve the equation x=\frac{3±3\sqrt{569}}{284} when ± is minus. Subtract 3\sqrt{569} from 3.

$$x=\frac{3-3\sqrt{569}}{284}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+3\sqrt{569}}{284} for x_{1} and \frac{3-3\sqrt{569}}{284} for x_{2}.

$$142x^{2}-3x-9=142\left(x-\frac{3\sqrt{569}+3}{284}\right)\left(x-\frac{3-3\sqrt{569}}{284}\right)$$

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 142

$$x ^ 2 -\frac{3}{142}x -\frac{9}{142} = 0$$

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

$$r + s = \frac{3}{142} rs = -\frac{9}{142}$$

Two numbers r and s sum up to \frac{3}{142} exactly when the average of the two numbers is \frac{1}{2}*\frac{3}{142} = \frac{3}{284}. 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>

$$r = \frac{3}{284} - u s = \frac{3}{284} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{9}{142}

$$(\frac{3}{284} - u) (\frac{3}{284} + u) = -\frac{9}{142}$$

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

$$\frac{9}{80656} - u^2 = -\frac{9}{142}$$

Simplify the expression by subtracting \frac{9}{80656} on both sides

$$-u^2 = -\frac{9}{142}-\frac{9}{80656} = -\frac{5121}{80656}$$

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

$$u^2 = \frac{5121}{80656} u = \pm\sqrt{\frac{5121}{80656}} = \pm \frac{\sqrt{5121}}{284}$$

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

$$r =\frac{3}{284} - \frac{\sqrt{5121}}{284} = -0.241 s = \frac{3}{284} + \frac{\sqrt{5121}}{284} = 0.263$$

Evaluate

$$142x^{2}-3x-9$$