Solve x^2+16x+1=0 | Microsoft Math Solver (2024)

x^{2}+16x+1=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{-16±\sqrt{16^{2}-4}}{2}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 16 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-16±\sqrt{256-4}}{2}

Square 16.

x=\frac{-16±\sqrt{252}}{2}

Add 256 to -4.

x=\frac{-16±6\sqrt{7}}{2}

Take the square root of 252.

x=\frac{6\sqrt{7}-16}{2}

Now solve the equation x=\frac{-16±6\sqrt{7}}{2} when ± is plus. Add -16 to 6\sqrt{7}.

x=3\sqrt{7}-8

Divide -16+6\sqrt{7} by 2.

x=\frac{-6\sqrt{7}-16}{2}

Now solve the equation x=\frac{-16±6\sqrt{7}}{2} when ± is minus. Subtract 6\sqrt{7} from -16.

x=-3\sqrt{7}-8

Divide -16-6\sqrt{7} by 2.

x=3\sqrt{7}-8 x=-3\sqrt{7}-8

The equation is now solved.

x^{2}+16x+1=0

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.

x^{2}+16x+1-1=-1

Subtract 1 from both sides of the equation.

x^{2}+16x=-1

Subtracting 1 from itself leaves 0.

x^{2}+16x+8^{2}=-1+8^{2}

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

x^{2}+16x+64=-1+64

Square 8.

x^{2}+16x+64=63

Add -1 to 64.

\left(x+8\right)^{2}=63

Factor x^{2}+16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

\sqrt{\left(x+8\right)^{2}}=\sqrt{63}

Take the square root of both sides of the equation.

x+8=3\sqrt{7} x+8=-3\sqrt{7}

Simplify.

x=3\sqrt{7}-8 x=-3\sqrt{7}-8

Subtract 8 from both sides of the equation.

x ^ 2 +16x +1 = 0

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.

r + s = -16 rs = 1

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 = -8 - u s = -8 + u

Two numbers r and s sum up to -16 exactly when the average of the two numbers is \frac{1}{2}*-16 = -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>

(-8 - u) (-8 + u) = 1

To solve for unknown quantity u, substitute these in the product equation rs = 1

64 - u^2 = 1

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

-u^2 = 1-64 = -63

Simplify the expression by subtracting 64 on both sides

u^2 = 63 u = \pm\sqrt{63} = \pm \sqrt{63}

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

r =-8 - \sqrt{63} = -15.937 s = -8 + \sqrt{63} = -0.063

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