Solve 37x^2-5x-24

$$37 x ^ { 2 } - 5 x - 24$$

Factor

$$37\left(x-\frac{5-7\sqrt{73}}{74}\right)\left(x-\frac{7\sqrt{73}+5}{74}\right)$$

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.

$$37x^{2}-5x-24=0$$

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.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 37\left(-24\right)}}{2\times 37}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 37\left(-24\right)}}{2\times 37}$$

Multiply -4 times 37.

$$x=\frac{-\left(-5\right)±\sqrt{25-148\left(-24\right)}}{2\times 37}$$

Multiply -148 times -24.

$$x=\frac{-\left(-5\right)±\sqrt{25+3552}}{2\times 37}$$

Add 25 to 3552.

$$x=\frac{-\left(-5\right)±\sqrt{3577}}{2\times 37}$$

Take the square root of 3577.

$$x=\frac{-\left(-5\right)±7\sqrt{73}}{2\times 37}$$

The opposite of -5 is 5.

$$x=\frac{5±7\sqrt{73}}{2\times 37}$$

Multiply 2 times 37.

$$x=\frac{5±7\sqrt{73}}{74}$$

Now solve the equation x=\frac{5±7\sqrt{73}}{74} when ± is plus. Add 5 to 7\sqrt{73}.

$$x=\frac{7\sqrt{73}+5}{74}$$

Now solve the equation x=\frac{5±7\sqrt{73}}{74} when ± is minus. Subtract 7\sqrt{73} from 5.

$$x=\frac{5-7\sqrt{73}}{74}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5+7\sqrt{73}}{74} for x_{1} and \frac{5-7\sqrt{73}}{74} for x_{2}.

$$37x^{2}-5x-24=37\left(x-\frac{7\sqrt{73}+5}{74}\right)\left(x-\frac{5-7\sqrt{73}}{74}\right)$$

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 37

$$x ^ 2 -\frac{5}{37}x -\frac{24}{37} = 0$$

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

$$r + s = \frac{5}{37} rs = -\frac{24}{37}$$

Two numbers r and s sum up to \frac{5}{37} exactly when the average of the two numbers is \frac{1}{2}*\frac{5}{37} = \frac{5}{74}. 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>

$$r = \frac{5}{74} - u s = \frac{5}{74} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{24}{37}

$$(\frac{5}{74} - u) (\frac{5}{74} + u) = -\frac{24}{37}$$

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

$$\frac{25}{5476} - u^2 = -\frac{24}{37}$$

Simplify the expression by subtracting \frac{25}{5476} on both sides

$$-u^2 = -\frac{24}{37}-\frac{25}{5476} = -\frac{3577}{5476}$$

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

$$u^2 = \frac{3577}{5476} u = \pm\sqrt{\frac{3577}{5476}} = \pm \frac{\sqrt{3577}}{74}$$

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

$$r =\frac{5}{74} - \frac{\sqrt{3577}}{74} = -0.741 s = \frac{5}{74} + \frac{\sqrt{3577}}{74} = 0.876$$

Evaluate

$$37x^{2}-5x-24$$