Solve 0x^2-25x-15

$$0 x ^ { 2 } - 25 x - 15$$

Evaluate

$$-25x-15$$

Solution Steps

Anything times zero gives zero.

$$0-25x-15$$

Subtract 15 from 0 to get -15.

$$-15-25x$$

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.

$$x ^ 2 -25x -15 = 0$$

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

$$r + s = 25 rs = -15$$

Two numbers r and s sum up to 25 exactly when the average of the two numbers is \frac{1}{2}*25 = \frac{25}{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>

$$r = \frac{25}{2} - u s = \frac{25}{2} + u$$

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

$$(\frac{25}{2} - u) (\frac{25}{2} + u) = -15$$

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

$$\frac{625}{4} - u^2 = -15$$

Simplify the expression by subtracting \frac{625}{4} on both sides

$$-u^2 = -15-\frac{625}{4} = -\frac{685}{4}$$

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

$$u^2 = \frac{685}{4} u = \pm\sqrt{\frac{685}{4}} = \pm \frac{\sqrt{685}}{2}$$

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

$$r =\frac{25}{2} - \frac{\sqrt{685}}{2} = -0.586 s = \frac{25}{2} + \frac{\sqrt{685}}{2} = 25.586$$

Factor

$$5\left(-5x-3\right)$$

Solution Steps

Factor out 5.

$$5\left(0-5x-3\right)$$