Solve 0a^2+14a-5=0

$$0 a ^ { 2 } + 14 a - 5 = 0$$

Solve for a

$$a=\frac{5}{14}\approx 0.357142857$$

Steps for Solving Linear Equation

Anything times zero gives zero.

$$0+14a-5=0$$

Subtract 5 from 0 to get -5.

$$-5+14a=0$$

Add 5 to both sides. Anything plus zero gives itself.

$$14a=5$$

Divide both sides by 14.

$$a=\frac{5}{14}$$

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.

$$x ^ 2 +14x -5 = 0$$

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

$$r + s = -14 rs = -5$$

Two numbers r and s sum up to -14 exactly when the average of the two numbers is \frac{1}{2}*-14 = -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>

$$r = -7 - u s = -7 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -5

$$(-7 - u) (-7 + u) = -5$$

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

$$49 - u^2 = -5$$

Simplify the expression by subtracting 49 on both sides

$$-u^2 = -5-49 = -54$$

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

$$u^2 = 54 u = \pm\sqrt{54} = \pm \sqrt{54}$$

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

$$r =-7 - \sqrt{54} = -14.348 s = -7 + \sqrt{54} = 0.348$$