Solve 16x^2-4x-21

$$16 x ^ { 2 } - 4 x - 21$$

Factor

$$16\left(x-\frac{1-\sqrt{85}}{8}\right)\left(x-\frac{\sqrt{85}+1}{8}\right)$$

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.

$$16x^{2}-4x-21=0$$

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.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 16\left(-21\right)}}{2\times 16}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 16\left(-21\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-4\right)±\sqrt{16-64\left(-21\right)}}{2\times 16}$$

Multiply -64 times -21.

$$x=\frac{-\left(-4\right)±\sqrt{16+1344}}{2\times 16}$$

Add 16 to 1344.

$$x=\frac{-\left(-4\right)±\sqrt{1360}}{2\times 16}$$

Take the square root of 1360.

$$x=\frac{-\left(-4\right)±4\sqrt{85}}{2\times 16}$$

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{85}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{4±4\sqrt{85}}{32}$$

Now solve the equation x=\frac{4±4\sqrt{85}}{32} when ± is plus. Add 4 to 4\sqrt{85}.

$$x=\frac{4\sqrt{85}+4}{32}$$

Divide 4+4\sqrt{85} by 32.

$$x=\frac{\sqrt{85}+1}{8}$$

Now solve the equation x=\frac{4±4\sqrt{85}}{32} when ± is minus. Subtract 4\sqrt{85} from 4.

$$x=\frac{4-4\sqrt{85}}{32}$$

Divide 4-4\sqrt{85} by 32.

$$x=\frac{1-\sqrt{85}}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{85}}{8} for x_{1} and \frac{1-\sqrt{85}}{8} for x_{2}.

$$16x^{2}-4x-21=16\left(x-\frac{\sqrt{85}+1}{8}\right)\left(x-\frac{1-\sqrt{85}}{8}\right)$$

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 16

$$x ^ 2 -\frac{1}{4}x -\frac{21}{16} = 0$$

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

$$r + s = \frac{1}{4} rs = -\frac{21}{16}$$

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

$$r = \frac{1}{8} - u s = \frac{1}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{21}{16}

$$(\frac{1}{8} - u) (\frac{1}{8} + u) = -\frac{21}{16}$$

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

$$\frac{1}{64} - u^2 = -\frac{21}{16}$$

Simplify the expression by subtracting \frac{1}{64} on both sides

$$-u^2 = -\frac{21}{16}-\frac{1}{64} = -\frac{85}{64}$$

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

$$u^2 = \frac{85}{64} u = \pm\sqrt{\frac{85}{64}} = \pm \frac{\sqrt{85}}{8}$$

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

$$r =\frac{1}{8} - \frac{\sqrt{85}}{8} = -1.027 s = \frac{1}{8} + \frac{\sqrt{85}}{8} = 1.277$$

Evaluate

$$16x^{2}-4x-21$$