Solve 15x^2+7x-72

$$15 x ^ { 2 } + 7 x - 72$$

Factor

$$15\left(x-\frac{-\sqrt{4369}-7}{30}\right)\left(x-\frac{\sqrt{4369}-7}{30}\right)$$

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.

$$15x^{2}+7x-72=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\times 15\left(-72\right)}}{2\times 15}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 15\left(-72\right)}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-7±\sqrt{49-60\left(-72\right)}}{2\times 15}$$

Multiply -60 times -72.

$$x=\frac{-7±\sqrt{49+4320}}{2\times 15}$$

Add 49 to 4320.

$$x=\frac{-7±\sqrt{4369}}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{-7±\sqrt{4369}}{30}$$

Now solve the equation x=\frac{-7±\sqrt{4369}}{30} when ± is plus. Add -7 to \sqrt{4369}.

$$x=\frac{\sqrt{4369}-7}{30}$$

Now solve the equation x=\frac{-7±\sqrt{4369}}{30} when ± is minus. Subtract \sqrt{4369} from -7.

$$x=\frac{-\sqrt{4369}-7}{30}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7+\sqrt{4369}}{30} for x_{1} and \frac{-7-\sqrt{4369}}{30} for x_{2}.

$$15x^{2}+7x-72=15\left(x-\frac{\sqrt{4369}-7}{30}\right)\left(x-\frac{-\sqrt{4369}-7}{30}\right)$$

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

$$x ^ 2 +\frac{7}{15}x -\frac{24}{5} = 0$$

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

$$r + s = -\frac{7}{15} rs = -\frac{24}{5}$$

Two numbers r and s sum up to -\frac{7}{15} exactly when the average of the two numbers is \frac{1}{2}*-\frac{7}{15} = -\frac{7}{30}. 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>

$$r = -\frac{7}{30} - u s = -\frac{7}{30} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{24}{5}

$$(-\frac{7}{30} - u) (-\frac{7}{30} + u) = -\frac{24}{5}$$

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

$$\frac{49}{900} - u^2 = -\frac{24}{5}$$

Simplify the expression by subtracting \frac{49}{900} on both sides

$$-u^2 = -\frac{24}{5}-\frac{49}{900} = -\frac{4369}{900}$$

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

$$u^2 = \frac{4369}{900} u = \pm\sqrt{\frac{4369}{900}} = \pm \frac{\sqrt{4369}}{30}$$

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

$$r =-\frac{7}{30} - \frac{\sqrt{4369}}{30} = -2.437 s = -\frac{7}{30} + \frac{\sqrt{4369}}{30} = 1.970$$

Evaluate

$$15x^{2}+7x-72$$