Solve 16x^2-224x-1561=0

$$16 x ^ { 2 } - 224 x - 1561 = 0$$

Solve for x

$$x=\frac{\sqrt{2345}}{4}+7\approx 19.106300013$$

$$x=-\frac{\sqrt{2345}}{4}+7\approx -5.106300013$$

Steps Using the Quadratic Formula

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.

$$16x^{2}-224x-1561=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -224 for b, and -1561 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-224\right)±\sqrt{\left(-224\right)^{2}-4\times 16\left(-1561\right)}}{2\times 16}$$

Square -224.

$$x=\frac{-\left(-224\right)±\sqrt{50176-4\times 16\left(-1561\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-224\right)±\sqrt{50176-64\left(-1561\right)}}{2\times 16}$$

Multiply -64 times -1561.

$$x=\frac{-\left(-224\right)±\sqrt{50176+99904}}{2\times 16}$$

Add 50176 to 99904.

$$x=\frac{-\left(-224\right)±\sqrt{150080}}{2\times 16}$$

Take the square root of 150080.

$$x=\frac{-\left(-224\right)±8\sqrt{2345}}{2\times 16}$$

The opposite of -224 is 224.

$$x=\frac{224±8\sqrt{2345}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{224±8\sqrt{2345}}{32}$$

Now solve the equation x=\frac{224±8\sqrt{2345}}{32} when ± is plus. Add 224 to 8\sqrt{2345}.

$$x=\frac{8\sqrt{2345}+224}{32}$$

Divide 224+8\sqrt{2345} by 32.

$$x=\frac{\sqrt{2345}}{4}+7$$

Now solve the equation x=\frac{224±8\sqrt{2345}}{32} when ± is minus. Subtract 8\sqrt{2345} from 224.

$$x=\frac{224-8\sqrt{2345}}{32}$$

Divide 224-8\sqrt{2345} by 32.

$$x=-\frac{\sqrt{2345}}{4}+7$$

The equation is now solved.

$$x=\frac{\sqrt{2345}}{4}+7 x=-\frac{\sqrt{2345}}{4}+7$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$16x^{2}-224x-1561=0$$

Add 1561 to both sides of the equation.

$$16x^{2}-224x-1561-\left(-1561\right)=-\left(-1561\right)$$

Subtracting -1561 from itself leaves 0.

$$16x^{2}-224x=-\left(-1561\right)$$

Subtract -1561 from 0.

$$16x^{2}-224x=1561$$

Divide both sides by 16.

$$\frac{16x^{2}-224x}{16}=\frac{1561}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{224}{16}\right)x=\frac{1561}{16}$$

Divide -224 by 16.

$$x^{2}-14x=\frac{1561}{16}$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=\frac{1561}{16}+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=\frac{1561}{16}+49$$

Add \frac{1561}{16} to 49.

$$x^{2}-14x+49=\frac{2345}{16}$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=\frac{2345}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{\frac{2345}{16}}$$

Simplify.

$$x-7=\frac{\sqrt{2345}}{4} x-7=-\frac{\sqrt{2345}}{4}$$

Add 7 to both sides of the equation.

$$x=\frac{\sqrt{2345}}{4}+7 x=-\frac{\sqrt{2345}}{4}+7$$

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

$$x ^ 2 -14x -\frac{1561}{16} = 0$$

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

$$r + s = 14 rs = -\frac{1561}{16}$$

Two numbers r and s sum up to 14 exactly when the average of the two numbers is \frac{1}{2}*14 = 7. 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>

$$r = 7 - u s = 7 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1561}{16}

$$(7 - u) (7 + u) = -\frac{1561}{16}$$

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

$$49 - u^2 = -\frac{1561}{16}$$

Simplify the expression by subtracting 49 on both sides

$$-u^2 = -\frac{1561}{16}-49 = -\frac{2345}{16}$$

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

$$u^2 = \frac{2345}{16} u = \pm\sqrt{\frac{2345}{16}} = \pm \frac{\sqrt{2345}}{4}$$

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

$$r =7 - \frac{\sqrt{2345}}{4} = -5.106 s = 7 + \frac{\sqrt{2345}}{4} = 19.106$$