Solve x^2-2x-800=0 | Microsoft Math Solver (2024)

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x^{2}-2x-800=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(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-800\right)}}{2}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-\left(-2\right)±\sqrt{4-4\left(-800\right)}}{2}

Square -2.

x=\frac{-\left(-2\right)±\sqrt{4+3200}}{2}

Multiply -4 times -800.

x=\frac{-\left(-2\right)±\sqrt{3204}}{2}

Add 4 to 3200.

x=\frac{-\left(-2\right)±6\sqrt{89}}{2}

Take the square root of 3204.

x=\frac{2±6\sqrt{89}}{2}

The opposite of -2 is 2.

x=\frac{6\sqrt{89}+2}{2}

Now solve the equation x=\frac{2±6\sqrt{89}}{2} when ± is plus. Add 2 to 6\sqrt{89}.

x=3\sqrt{89}+1

Divide 2+6\sqrt{89} by 2.

x=\frac{2-6\sqrt{89}}{2}

Now solve the equation x=\frac{2±6\sqrt{89}}{2} when ± is minus. Subtract 6\sqrt{89} from 2.

x=1-3\sqrt{89}

Divide 2-6\sqrt{89} by 2.

x=3\sqrt{89}+1 x=1-3\sqrt{89}

The equation is now solved.

x^{2}-2x-800=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}-2x-800-\left(-800\right)=-\left(-800\right)

Add 800 to both sides of the equation.

x^{2}-2x=-\left(-800\right)

Subtracting -800 from itself leaves 0.

x^{2}-2x=800

Subtract -800 from 0.

x^{2}-2x+1=800+1

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

x^{2}-2x+1=801

Add 800 to 1.

\left(x-1\right)^{2}=801

Factor x^{2}-2x+1. 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-1\right)^{2}}=\sqrt{801}

Take the square root of both sides of the equation.

x-1=3\sqrt{89} x-1=-3\sqrt{89}

Simplify.

x=3\sqrt{89}+1 x=1-3\sqrt{89}

Add 1 to both sides of the equation.

x ^ 2 -2x -800 = 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 = 2 rs = -800

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 = 1 - u s = 1 + u

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

(1 - u) (1 + u) = -800

To solve for unknown quantity u, substitute these in the product equation rs = -800

1 - u^2 = -800

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

-u^2 = -800-1 = -801

Simplify the expression by subtracting 1 on both sides

u^2 = 801 u = \pm\sqrt{801} = \pm \sqrt{801}

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

r =1 - \sqrt{801} = -27.302 s = 1 + \sqrt{801} = 29.302

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