Solve 15x^2-82x+24

$$15 x ^ { 2 } - 82 x + 24$$

Factor

$$15\left(x-\frac{41-\sqrt{1321}}{15}\right)\left(x-\frac{\sqrt{1321}+41}{15}\right)$$

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.

$$15x^{2}-82x+24=0$$

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.

$$x=\frac{-\left(-82\right)±\sqrt{\left(-82\right)^{2}-4\times 15\times 24}}{2\times 15}$$

Square -82.

$$x=\frac{-\left(-82\right)±\sqrt{6724-4\times 15\times 24}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-\left(-82\right)±\sqrt{6724-60\times 24}}{2\times 15}$$

Multiply -60 times 24.

$$x=\frac{-\left(-82\right)±\sqrt{6724-1440}}{2\times 15}$$

Add 6724 to -1440.

$$x=\frac{-\left(-82\right)±\sqrt{5284}}{2\times 15}$$

Take the square root of 5284.

$$x=\frac{-\left(-82\right)±2\sqrt{1321}}{2\times 15}$$

The opposite of -82 is 82.

$$x=\frac{82±2\sqrt{1321}}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{82±2\sqrt{1321}}{30}$$

Now solve the equation x=\frac{82±2\sqrt{1321}}{30} when ± is plus. Add 82 to 2\sqrt{1321}.

$$x=\frac{2\sqrt{1321}+82}{30}$$

Divide 82+2\sqrt{1321} by 30.

$$x=\frac{\sqrt{1321}+41}{15}$$

Now solve the equation x=\frac{82±2\sqrt{1321}}{30} when ± is minus. Subtract 2\sqrt{1321} from 82.

$$x=\frac{82-2\sqrt{1321}}{30}$$

Divide 82-2\sqrt{1321} by 30.

$$x=\frac{41-\sqrt{1321}}{15}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{41+\sqrt{1321}}{15} for x_{1} and \frac{41-\sqrt{1321}}{15} for x_{2}.

$$15x^{2}-82x+24=15\left(x-\frac{\sqrt{1321}+41}{15}\right)\left(x-\frac{41-\sqrt{1321}}{15}\right)$$

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 15

$$x ^ 2 -\frac{82}{15}x +\frac{8}{5} = 0$$

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

$$r + s = \frac{82}{15} rs = \frac{8}{5}$$

Two numbers r and s sum up to \frac{82}{15} exactly when the average of the two numbers is \frac{1}{2}*\frac{82}{15} = \frac{41}{15}. 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>

$$r = \frac{41}{15} - u s = \frac{41}{15} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{8}{5}

$$(\frac{41}{15} - u) (\frac{41}{15} + u) = \frac{8}{5}$$

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

$$\frac{1681}{225} - u^2 = \frac{8}{5}$$

Simplify the expression by subtracting \frac{1681}{225} on both sides

$$-u^2 = \frac{8}{5}-\frac{1681}{225} = -\frac{1321}{225}$$

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

$$u^2 = \frac{1321}{225} u = \pm\sqrt{\frac{1321}{225}} = \pm \frac{\sqrt{1321}}{15}$$

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

$$r =\frac{41}{15} - \frac{\sqrt{1321}}{15} = 0.310 s = \frac{41}{15} + \frac{\sqrt{1321}}{15} = 5.156$$

Evaluate

$$15x^{2}-82x+24$$