Solve 4a^2-16a-27

$$4 a ^ { 2 } - 16 a - 27$$

Factor

$$4\left(a-\left(-\frac{\sqrt{43}}{2}+2\right)\right)\left(a-\left(\frac{\sqrt{43}}{2}+2\right)\right)$$

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.

$$4a^{2}-16a-27=0$$

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.

$$a=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 4\left(-27\right)}}{2\times 4}$$

Square -16.

$$a=\frac{-\left(-16\right)±\sqrt{256-4\times 4\left(-27\right)}}{2\times 4}$$

Multiply -4 times 4.

$$a=\frac{-\left(-16\right)±\sqrt{256-16\left(-27\right)}}{2\times 4}$$

Multiply -16 times -27.

$$a=\frac{-\left(-16\right)±\sqrt{256+432}}{2\times 4}$$

Add 256 to 432.

$$a=\frac{-\left(-16\right)±\sqrt{688}}{2\times 4}$$

Take the square root of 688.

$$a=\frac{-\left(-16\right)±4\sqrt{43}}{2\times 4}$$

The opposite of -16 is 16.

$$a=\frac{16±4\sqrt{43}}{2\times 4}$$

Multiply 2 times 4.

$$a=\frac{16±4\sqrt{43}}{8}$$

Now solve the equation a=\frac{16±4\sqrt{43}}{8} when ± is plus. Add 16 to 4\sqrt{43}.

$$a=\frac{4\sqrt{43}+16}{8}$$

Divide 16+4\sqrt{43} by 8.

$$a=\frac{\sqrt{43}}{2}+2$$

Now solve the equation a=\frac{16±4\sqrt{43}}{8} when ± is minus. Subtract 4\sqrt{43} from 16.

$$a=\frac{16-4\sqrt{43}}{8}$$

Divide 16-4\sqrt{43} by 8.

$$a=-\frac{\sqrt{43}}{2}+2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2+\frac{\sqrt{43}}{2} for x_{1} and 2-\frac{\sqrt{43}}{2} for x_{2}.

$$4a^{2}-16a-27=4\left(a-\left(\frac{\sqrt{43}}{2}+2\right)\right)\left(a-\left(-\frac{\sqrt{43}}{2}+2\right)\right)$$

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 4

$$x ^ 2 -4x -\frac{27}{4} = 0$$

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

$$r + s = 4 rs = -\frac{27}{4}$$

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

$$r = 2 - u s = 2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{27}{4}

$$(2 - u) (2 + u) = -\frac{27}{4}$$

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

$$4 - u^2 = -\frac{27}{4}$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -\frac{27}{4}-4 = -\frac{43}{4}$$

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

$$u^2 = \frac{43}{4} u = \pm\sqrt{\frac{43}{4}} = \pm \frac{\sqrt{43}}{2}$$

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

$$r =2 - \frac{\sqrt{43}}{2} = -1.279 s = 2 + \frac{\sqrt{43}}{2} = 5.279$$

Evaluate

$$4a^{2}-16a-27$$