Solve quad205x^2+32x-21

$$\quad 205 x ^ { 2 } + 32 x - 21$$

Factor

$$205\left(x-\frac{-\sqrt{4561}-16}{205}\right)\left(x-\frac{\sqrt{4561}-16}{205}\right)$$

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.

$$205x^{2}+32x-21=0$$

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.

$$x=\frac{-32±\sqrt{32^{2}-4\times 205\left(-21\right)}}{2\times 205}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\times 205\left(-21\right)}}{2\times 205}$$

Multiply -4 times 205.

$$x=\frac{-32±\sqrt{1024-820\left(-21\right)}}{2\times 205}$$

Multiply -820 times -21.

$$x=\frac{-32±\sqrt{1024+17220}}{2\times 205}$$

Add 1024 to 17220.

$$x=\frac{-32±\sqrt{18244}}{2\times 205}$$

Take the square root of 18244.

$$x=\frac{-32±2\sqrt{4561}}{2\times 205}$$

Multiply 2 times 205.

$$x=\frac{-32±2\sqrt{4561}}{410}$$

Now solve the equation x=\frac{-32±2\sqrt{4561}}{410} when ± is plus. Add -32 to 2\sqrt{4561}.

$$x=\frac{2\sqrt{4561}-32}{410}$$

Divide -32+2\sqrt{4561} by 410.

$$x=\frac{\sqrt{4561}-16}{205}$$

Now solve the equation x=\frac{-32±2\sqrt{4561}}{410} when ± is minus. Subtract 2\sqrt{4561} from -32.

$$x=\frac{-2\sqrt{4561}-32}{410}$$

Divide -32-2\sqrt{4561} by 410.

$$x=\frac{-\sqrt{4561}-16}{205}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-16+\sqrt{4561}}{205} for x_{1} and \frac{-16-\sqrt{4561}}{205} for x_{2}.

$$205x^{2}+32x-21=205\left(x-\frac{\sqrt{4561}-16}{205}\right)\left(x-\frac{-\sqrt{4561}-16}{205}\right)$$

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 205

$$x ^ 2 +\frac{32}{205}x -\frac{21}{205} = 0$$

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

$$r + s = -\frac{32}{205} rs = -\frac{21}{205}$$

Two numbers r and s sum up to -\frac{32}{205} exactly when the average of the two numbers is \frac{1}{2}*-\frac{32}{205} = -\frac{16}{205}. 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>

$$r = -\frac{16}{205} - u s = -\frac{16}{205} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{21}{205}

$$(-\frac{16}{205} - u) (-\frac{16}{205} + u) = -\frac{21}{205}$$

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

$$\frac{256}{42025} - u^2 = -\frac{21}{205}$$

Simplify the expression by subtracting \frac{256}{42025} on both sides

$$-u^2 = -\frac{21}{205}-\frac{256}{42025} = -\frac{4561}{42025}$$

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

$$u^2 = \frac{4561}{42025} u = \pm\sqrt{\frac{4561}{42025}} = \pm \frac{\sqrt{4561}}{205}$$

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

$$r =-\frac{16}{205} - \frac{\sqrt{4561}}{205} = -0.407 s = -\frac{16}{205} + \frac{\sqrt{4561}}{205} = 0.251$$

Evaluate

$$205x^{2}+32x-21$$