Solve 5x^2-20x-75

$$5 x ^ { 2 } - 20 x - 75$$

Factor

$$5\left(x-\left(2-\sqrt{19}\right)\right)\left(x-\left(\sqrt{19}+2\right)\right)$$

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.

$$5x^{2}-20x-75=0$$

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.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 5\left(-75\right)}}{2\times 5}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 5\left(-75\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-20\right)±\sqrt{400-20\left(-75\right)}}{2\times 5}$$

Multiply -20 times -75.

$$x=\frac{-\left(-20\right)±\sqrt{400+1500}}{2\times 5}$$

Add 400 to 1500.

$$x=\frac{-\left(-20\right)±\sqrt{1900}}{2\times 5}$$

Take the square root of 1900.

$$x=\frac{-\left(-20\right)±10\sqrt{19}}{2\times 5}$$

The opposite of -20 is 20.

$$x=\frac{20±10\sqrt{19}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{20±10\sqrt{19}}{10}$$

Now solve the equation x=\frac{20±10\sqrt{19}}{10} when ± is plus. Add 20 to 10\sqrt{19}.

$$x=\frac{10\sqrt{19}+20}{10}$$

Divide 20+10\sqrt{19} by 10.

$$x=\sqrt{19}+2$$

Now solve the equation x=\frac{20±10\sqrt{19}}{10} when ± is minus. Subtract 10\sqrt{19} from 20.

$$x=\frac{20-10\sqrt{19}}{10}$$

Divide 20-10\sqrt{19} by 10.

$$x=2-\sqrt{19}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2+\sqrt{19} for x_{1} and 2-\sqrt{19} for x_{2}.

$$5x^{2}-20x-75=5\left(x-\left(\sqrt{19}+2\right)\right)\left(x-\left(2-\sqrt{19}\right)\right)$$

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.This is achieved by dividing both sides of the equation by 5

$$x ^ 2 -4x -15 = 0$$

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

$$r + s = 4 rs = -15$$

Two numbers r and s sum up to 4 exactly when the average of the two numbers is \frac{1}{2}*4 = 2. 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>

$$r = 2 - u s = 2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -15

$$(2 - u) (2 + u) = -15$$

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

$$4 - u^2 = -15$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -15-4 = -19$$

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

$$u^2 = 19 u = \pm\sqrt{19} = \pm \sqrt{19}$$

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

$$r =2 - \sqrt{19} = -2.359 s = 2 + \sqrt{19} = 6.359$$

Evaluate

$$5\left(x^{2}-4x-15\right)$$