Solve 700x^2+9200x-1800

$$700 x ^ { 2 } + 9200 x - 1800$$

Factor

$$700\left(x-\frac{-\sqrt{2242}-46}{7}\right)\left(x-\frac{\sqrt{2242}-46}{7}\right)$$

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.

$$700x^{2}+9200x-1800=0$$

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.

$$x=\frac{-9200±\sqrt{9200^{2}-4\times 700\left(-1800\right)}}{2\times 700}$$

Square 9200.

$$x=\frac{-9200±\sqrt{84640000-4\times 700\left(-1800\right)}}{2\times 700}$$

Multiply -4 times 700.

$$x=\frac{-9200±\sqrt{84640000-2800\left(-1800\right)}}{2\times 700}$$

Multiply -2800 times -1800.

$$x=\frac{-9200±\sqrt{84640000+5040000}}{2\times 700}$$

Add 84640000 to 5040000.

$$x=\frac{-9200±\sqrt{89680000}}{2\times 700}$$

Take the square root of 89680000.

$$x=\frac{-9200±200\sqrt{2242}}{2\times 700}$$

Multiply 2 times 700.

$$x=\frac{-9200±200\sqrt{2242}}{1400}$$

Now solve the equation x=\frac{-9200±200\sqrt{2242}}{1400} when ± is plus. Add -9200 to 200\sqrt{2242}.

$$x=\frac{200\sqrt{2242}-9200}{1400}$$

Divide -9200+200\sqrt{2242} by 1400.

$$x=\frac{\sqrt{2242}-46}{7}$$

Now solve the equation x=\frac{-9200±200\sqrt{2242}}{1400} when ± is minus. Subtract 200\sqrt{2242} from -9200.

$$x=\frac{-200\sqrt{2242}-9200}{1400}$$

Divide -9200-200\sqrt{2242} by 1400.

$$x=\frac{-\sqrt{2242}-46}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-46+\sqrt{2242}}{7} for x_{1} and \frac{-46-\sqrt{2242}}{7} for x_{2}.

$$700x^{2}+9200x-1800=700\left(x-\frac{\sqrt{2242}-46}{7}\right)\left(x-\frac{-\sqrt{2242}-46}{7}\right)$$

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 700

$$x ^ 2 +\frac{92}{7}x -\frac{18}{7} = 0$$

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

$$r + s = -\frac{92}{7} rs = -\frac{18}{7}$$

Two numbers r and s sum up to -\frac{92}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{92}{7} = -\frac{46}{7}. 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>

$$r = -\frac{46}{7} - u s = -\frac{46}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{18}{7}

$$(-\frac{46}{7} - u) (-\frac{46}{7} + u) = -\frac{18}{7}$$

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

$$\frac{2116}{49} - u^2 = -\frac{18}{7}$$

Simplify the expression by subtracting \frac{2116}{49} on both sides

$$-u^2 = -\frac{18}{7}-\frac{2116}{49} = -\frac{2242}{49}$$

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

$$u^2 = \frac{2242}{49} u = \pm\sqrt{\frac{2242}{49}} = \pm \frac{\sqrt{2242}}{7}$$

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

$$r =-\frac{46}{7} - \frac{\sqrt{2242}}{7} = -13.336 s = -\frac{46}{7} + \frac{\sqrt{2242}}{7} = 0.193$$

Evaluate

$$700x^{2}+9200x-1800$$