Solve -4x^2-16x+3

$$- 4 x ^ { 2 } - 16 x + 3$$

Factor

$$-4\left(x-\left(-\frac{\sqrt{19}}{2}-2\right)\right)\left(x-\left(\frac{\sqrt{19}}{2}-2\right)\right)$$

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.

$$-4x^{2}-16x+3=0$$

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.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-4\right)\times 3}}{2\left(-4\right)}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-4\right)\times 3}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-16\right)±\sqrt{256+16\times 3}}{2\left(-4\right)}$$

Multiply 16 times 3.

$$x=\frac{-\left(-16\right)±\sqrt{256+48}}{2\left(-4\right)}$$

Add 256 to 48.

$$x=\frac{-\left(-16\right)±\sqrt{304}}{2\left(-4\right)}$$

Take the square root of 304.

$$x=\frac{-\left(-16\right)±4\sqrt{19}}{2\left(-4\right)}$$

The opposite of -16 is 16.

$$x=\frac{16±4\sqrt{19}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{16±4\sqrt{19}}{-8}$$

Now solve the equation x=\frac{16±4\sqrt{19}}{-8} when ± is plus. Add 16 to 4\sqrt{19}.

$$x=\frac{4\sqrt{19}+16}{-8}$$

Divide 16+4\sqrt{19} by -8.

$$x=-\frac{\sqrt{19}}{2}-2$$

Now solve the equation x=\frac{16±4\sqrt{19}}{-8} when ± is minus. Subtract 4\sqrt{19} from 16.

$$x=\frac{16-4\sqrt{19}}{-8}$$

Divide 16-4\sqrt{19} by -8.

$$x=\frac{\sqrt{19}}{2}-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2-\frac{\sqrt{19}}{2} for x_{1} and -2+\frac{\sqrt{19}}{2} for x_{2}.

$$-4x^{2}-16x+3=-4\left(x-\left(-\frac{\sqrt{19}}{2}-2\right)\right)\left(x-\left(\frac{\sqrt{19}}{2}-2\right)\right)$$

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.

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

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

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

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{3}{4}

$$(-2 - u) (-2 + u) = -\frac{3}{4}$$

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

$$4 - u^2 = -\frac{3}{4}$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -\frac{3}{4}-4 = -\frac{19}{4}$$

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

$$u^2 = \frac{19}{4} u = \pm\sqrt{\frac{19}{4}} = \pm \frac{\sqrt{19}}{2}$$

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

$$r =-2 - \frac{\sqrt{19}}{2} = -4.179 s = -2 + \frac{\sqrt{19}}{2} = 0.179$$

Evaluate

$$3-16x-4x^{2}$$