Solve 64h^2-48h-9

$$64 h ^ { 2 } - 48 h - 9$$

Factor

$$64\left(h-\frac{3-3\sqrt{2}}{8}\right)\left(h-\frac{3\sqrt{2}+3}{8}\right)$$

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.

$$64h^{2}-48h-9=0$$

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.

$$h=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 64\left(-9\right)}}{2\times 64}$$

Square -48.

$$h=\frac{-\left(-48\right)±\sqrt{2304-4\times 64\left(-9\right)}}{2\times 64}$$

Multiply -4 times 64.

$$h=\frac{-\left(-48\right)±\sqrt{2304-256\left(-9\right)}}{2\times 64}$$

Multiply -256 times -9.

$$h=\frac{-\left(-48\right)±\sqrt{2304+2304}}{2\times 64}$$

Add 2304 to 2304.

$$h=\frac{-\left(-48\right)±\sqrt{4608}}{2\times 64}$$

Take the square root of 4608.

$$h=\frac{-\left(-48\right)±48\sqrt{2}}{2\times 64}$$

The opposite of -48 is 48.

$$h=\frac{48±48\sqrt{2}}{2\times 64}$$

Multiply 2 times 64.

$$h=\frac{48±48\sqrt{2}}{128}$$

Now solve the equation h=\frac{48±48\sqrt{2}}{128} when ± is plus. Add 48 to 48\sqrt{2}.

$$h=\frac{48\sqrt{2}+48}{128}$$

Divide 48+48\sqrt{2} by 128.

$$h=\frac{3\sqrt{2}+3}{8}$$

Now solve the equation h=\frac{48±48\sqrt{2}}{128} when ± is minus. Subtract 48\sqrt{2} from 48.

$$h=\frac{48-48\sqrt{2}}{128}$$

Divide 48-48\sqrt{2} by 128.

$$h=\frac{3-3\sqrt{2}}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+3\sqrt{2}}{8} for x_{1} and \frac{3-3\sqrt{2}}{8} for x_{2}.

$$64h^{2}-48h-9=64\left(h-\frac{3\sqrt{2}+3}{8}\right)\left(h-\frac{3-3\sqrt{2}}{8}\right)$$

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 64

$$x ^ 2 -\frac{3}{4}x -\frac{9}{64} = 0$$

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

$$r + s = \frac{3}{4} rs = -\frac{9}{64}$$

Two numbers r and s sum up to \frac{3}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{3}{4} = \frac{3}{8}. 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>

$$r = \frac{3}{8} - u s = \frac{3}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{9}{64}

$$(\frac{3}{8} - u) (\frac{3}{8} + u) = -\frac{9}{64}$$

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

$$\frac{9}{64} - u^2 = -\frac{9}{64}$$

Simplify the expression by subtracting \frac{9}{64} on both sides

$$-u^2 = -\frac{9}{64}-\frac{9}{64} = -\frac{9}{32}$$

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

$$u^2 = \frac{9}{32} u = \pm\sqrt{\frac{9}{32}} = \pm \frac{3}{\sqrt{32}}$$

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

$$r =\frac{3}{8} - \frac{3}{\sqrt{32}} = -0.155 s = \frac{3}{8} + \frac{3}{\sqrt{32}} = 0.905$$

Evaluate

$$64h^{2}-48h-9$$