Solve 115a^2-5a-60

$$115 a ^ { 2 } - 5 a - 60$$

Factor

$$115\left(a-\frac{1-\sqrt{1105}}{46}\right)\left(a-\frac{\sqrt{1105}+1}{46}\right)$$

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.

$$115a^{2}-5a-60=0$$

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.

$$a=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 115\left(-60\right)}}{2\times 115}$$

Square -5.

$$a=\frac{-\left(-5\right)±\sqrt{25-4\times 115\left(-60\right)}}{2\times 115}$$

Multiply -4 times 115.

$$a=\frac{-\left(-5\right)±\sqrt{25-460\left(-60\right)}}{2\times 115}$$

Multiply -460 times -60.

$$a=\frac{-\left(-5\right)±\sqrt{25+27600}}{2\times 115}$$

Add 25 to 27600.

$$a=\frac{-\left(-5\right)±\sqrt{27625}}{2\times 115}$$

Take the square root of 27625.

$$a=\frac{-\left(-5\right)±5\sqrt{1105}}{2\times 115}$$

The opposite of -5 is 5.

$$a=\frac{5±5\sqrt{1105}}{2\times 115}$$

Multiply 2 times 115.

$$a=\frac{5±5\sqrt{1105}}{230}$$

Now solve the equation a=\frac{5±5\sqrt{1105}}{230} when ± is plus. Add 5 to 5\sqrt{1105}.

$$a=\frac{5\sqrt{1105}+5}{230}$$

Divide 5+5\sqrt{1105} by 230.

$$a=\frac{\sqrt{1105}+1}{46}$$

Now solve the equation a=\frac{5±5\sqrt{1105}}{230} when ± is minus. Subtract 5\sqrt{1105} from 5.

$$a=\frac{5-5\sqrt{1105}}{230}$$

Divide 5-5\sqrt{1105} by 230.

$$a=\frac{1-\sqrt{1105}}{46}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{1105}}{46} for x_{1} and \frac{1-\sqrt{1105}}{46} for x_{2}.

$$115a^{2}-5a-60=115\left(a-\frac{\sqrt{1105}+1}{46}\right)\left(a-\frac{1-\sqrt{1105}}{46}\right)$$

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 115

$$x ^ 2 -\frac{1}{23}x -\frac{12}{23} = 0$$

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

$$r + s = \frac{1}{23} rs = -\frac{12}{23}$$

Two numbers r and s sum up to \frac{1}{23} exactly when the average of the two numbers is \frac{1}{2}*\frac{1}{23} = \frac{1}{46}. 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>

$$r = \frac{1}{46} - u s = \frac{1}{46} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{12}{23}

$$(\frac{1}{46} - u) (\frac{1}{46} + u) = -\frac{12}{23}$$

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

$$\frac{1}{2116} - u^2 = -\frac{12}{23}$$

Simplify the expression by subtracting \frac{1}{2116} on both sides

$$-u^2 = -\frac{12}{23}-\frac{1}{2116} = -\frac{1105}{2116}$$

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

$$u^2 = \frac{1105}{2116} u = \pm\sqrt{\frac{1105}{2116}} = \pm \frac{\sqrt{1105}}{46}$$

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

$$r =\frac{1}{46} - \frac{\sqrt{1105}}{46} = -0.701 s = \frac{1}{46} + \frac{\sqrt{1105}}{46} = 0.744$$

Evaluate

$$115a^{2}-5a-60$$