Solve -5x^2+700x-6420

$$- 5 x ^ { 2 } + 700 x - 6420$$

Factor

$$-5\left(x-\left(70-4\sqrt{226}\right)\right)\left(x-\left(4\sqrt{226}+70\right)\right)$$

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.

$$-5x^{2}+700x-6420=0$$

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.

$$x=\frac{-700±\sqrt{700^{2}-4\left(-5\right)\left(-6420\right)}}{2\left(-5\right)}$$

Square 700.

$$x=\frac{-700±\sqrt{490000-4\left(-5\right)\left(-6420\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-700±\sqrt{490000+20\left(-6420\right)}}{2\left(-5\right)}$$

Multiply 20 times -6420.

$$x=\frac{-700±\sqrt{490000-128400}}{2\left(-5\right)}$$

Add 490000 to -128400.

$$x=\frac{-700±\sqrt{361600}}{2\left(-5\right)}$$

Take the square root of 361600.

$$x=\frac{-700±40\sqrt{226}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-700±40\sqrt{226}}{-10}$$

Now solve the equation x=\frac{-700±40\sqrt{226}}{-10} when ± is plus. Add -700 to 40\sqrt{226}.

$$x=\frac{40\sqrt{226}-700}{-10}$$

Divide -700+40\sqrt{226} by -10.

$$x=70-4\sqrt{226}$$

Now solve the equation x=\frac{-700±40\sqrt{226}}{-10} when ± is minus. Subtract 40\sqrt{226} from -700.

$$x=\frac{-40\sqrt{226}-700}{-10}$$

Divide -700-40\sqrt{226} by -10.

$$x=4\sqrt{226}+70$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 70-4\sqrt{226} for x_{1} and 70+4\sqrt{226} for x_{2}.

$$-5x^{2}+700x-6420=-5\left(x-\left(70-4\sqrt{226}\right)\right)\left(x-\left(4\sqrt{226}+70\right)\right)$$

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.

$$x ^ 2 -140x +1284 = 0$$

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

$$r + s = 140 rs = 1284$$

Two numbers r and s sum up to 140 exactly when the average of the two numbers is \frac{1}{2}*140 = 70. 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>

$$r = 70 - u s = 70 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 1284

$$(70 - u) (70 + u) = 1284$$

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

$$4900 - u^2 = 1284$$

Simplify the expression by subtracting 4900 on both sides

$$-u^2 = 1284-4900 = -3616$$

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

$$u^2 = 3616 u = \pm\sqrt{3616} = \pm \sqrt{3616}$$

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

$$r =70 - \sqrt{3616} = 9.867 s = 70 + \sqrt{3616} = 130.133$$

Evaluate

$$-5x^{2}+700x-6420$$