Solve 32x^2-5x-7

$$32 x ^ { 2 } - 5 x - 7$$

Factor

$$32\left(x-\frac{5-\sqrt{921}}{64}\right)\left(x-\frac{\sqrt{921}+5}{64}\right)$$

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.

$$32x^{2}-5x-7=0$$

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.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 32\left(-7\right)}}{2\times 32}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 32\left(-7\right)}}{2\times 32}$$

Multiply -4 times 32.

$$x=\frac{-\left(-5\right)±\sqrt{25-128\left(-7\right)}}{2\times 32}$$

Multiply -128 times -7.

$$x=\frac{-\left(-5\right)±\sqrt{25+896}}{2\times 32}$$

Add 25 to 896.

$$x=\frac{-\left(-5\right)±\sqrt{921}}{2\times 32}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{921}}{2\times 32}$$

Multiply 2 times 32.

$$x=\frac{5±\sqrt{921}}{64}$$

Now solve the equation x=\frac{5±\sqrt{921}}{64} when ± is plus. Add 5 to \sqrt{921}.

$$x=\frac{\sqrt{921}+5}{64}$$

Now solve the equation x=\frac{5±\sqrt{921}}{64} when ± is minus. Subtract \sqrt{921} from 5.

$$x=\frac{5-\sqrt{921}}{64}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5+\sqrt{921}}{64} for x_{1} and \frac{5-\sqrt{921}}{64} for x_{2}.

$$32x^{2}-5x-7=32\left(x-\frac{\sqrt{921}+5}{64}\right)\left(x-\frac{5-\sqrt{921}}{64}\right)$$

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 32

$$x ^ 2 -\frac{5}{32}x -\frac{7}{32} = 0$$

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

$$r + s = \frac{5}{32} rs = -\frac{7}{32}$$

Two numbers r and s sum up to \frac{5}{32} exactly when the average of the two numbers is \frac{1}{2}*\frac{5}{32} = \frac{5}{64}. 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>

$$r = \frac{5}{64} - u s = \frac{5}{64} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{7}{32}

$$(\frac{5}{64} - u) (\frac{5}{64} + u) = -\frac{7}{32}$$

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

$$\frac{25}{4096} - u^2 = -\frac{7}{32}$$

Simplify the expression by subtracting \frac{25}{4096} on both sides

$$-u^2 = -\frac{7}{32}-\frac{25}{4096} = -\frac{921}{4096}$$

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

$$u^2 = \frac{921}{4096} u = \pm\sqrt{\frac{921}{4096}} = \pm \frac{\sqrt{921}}{64}$$

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

$$r =\frac{5}{64} - \frac{\sqrt{921}}{64} = -0.396 s = \frac{5}{64} + \frac{\sqrt{921}}{64} = 0.552$$

Evaluate

$$32x^{2}-5x-7$$