Solve 146x^2+x-2

$$146 x ^ { 2 } + x - 2$$

Factor

$$146\left(x-\frac{-\sqrt{1169}-1}{292}\right)\left(x-\frac{\sqrt{1169}-1}{292}\right)$$

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.

$$146x^{2}+x-2=0$$

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.

$$x=\frac{-1±\sqrt{1^{2}-4\times 146\left(-2\right)}}{2\times 146}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 146\left(-2\right)}}{2\times 146}$$

Multiply -4 times 146.

$$x=\frac{-1±\sqrt{1-584\left(-2\right)}}{2\times 146}$$

Multiply -584 times -2.

$$x=\frac{-1±\sqrt{1+1168}}{2\times 146}$$

Add 1 to 1168.

$$x=\frac{-1±\sqrt{1169}}{2\times 146}$$

Multiply 2 times 146.

$$x=\frac{-1±\sqrt{1169}}{292}$$

Now solve the equation x=\frac{-1±\sqrt{1169}}{292} when ± is plus. Add -1 to \sqrt{1169}.

$$x=\frac{\sqrt{1169}-1}{292}$$

Now solve the equation x=\frac{-1±\sqrt{1169}}{292} when ± is minus. Subtract \sqrt{1169} from -1.

$$x=\frac{-\sqrt{1169}-1}{292}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1+\sqrt{1169}}{292} for x_{1} and \frac{-1-\sqrt{1169}}{292} for x_{2}.

$$146x^{2}+x-2=146\left(x-\frac{\sqrt{1169}-1}{292}\right)\left(x-\frac{-\sqrt{1169}-1}{292}\right)$$

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 146

$$x ^ 2 +\frac{1}{146}x -\frac{1}{73} = 0$$

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

$$r + s = -\frac{1}{146} rs = -\frac{1}{73}$$

Two numbers r and s sum up to -\frac{1}{146} exactly when the average of the two numbers is \frac{1}{2}*-\frac{1}{146} = -\frac{1}{292}. 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>

$$r = -\frac{1}{292} - u s = -\frac{1}{292} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{73}

$$(-\frac{1}{292} - u) (-\frac{1}{292} + u) = -\frac{1}{73}$$

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

$$\frac{1}{85264} - u^2 = -\frac{1}{73}$$

Simplify the expression by subtracting \frac{1}{85264} on both sides

$$-u^2 = -\frac{1}{73}-\frac{1}{85264} = -\frac{1169}{85264}$$

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

$$u^2 = \frac{1169}{85264} u = \pm\sqrt{\frac{1169}{85264}} = \pm \frac{\sqrt{1169}}{292}$$

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

$$r =-\frac{1}{292} - \frac{\sqrt{1169}}{292} = -0.121 s = -\frac{1}{292} + \frac{\sqrt{1169}}{292} = 0.114$$

Evaluate

$$146x^{2}+x-2$$