Solve x^2+100x+2500=0

$$x ^ { 2 } + 100 x + 2500 = 0$$

Solve for x

$$x=-50$$

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.

$$x^{2}+100x+2500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 100 for b, and 2500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\times 2500}}{2}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\times 2500}}{2}$$

Multiply -4 times 2500.

$$x=\frac{-100±\sqrt{10000-10000}}{2}$$

Add 10000 to -10000.

$$x=\frac{-100±\sqrt{0}}{2}$$

Take the square root of 0.

$$x=-\frac{100}{2}$$

Divide -100 by 2.

$$x=-50$$

Steps for Completing the Square

Factor x^{2}+100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+50\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+50\right)^{2}}=\sqrt{0}$$

Simplify.

$$x+50=0 x+50=0$$

Subtract 50 from both sides of the equation.

$$x=-50 x=-50$$

The equation is now solved. Solutions are the same.

$$x=-50$$

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.

$$x ^ 2 +100x +2500 = 0$$

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

$$r + s = -100 rs = 2500$$

Two numbers r and s sum up to -100 exactly when the average of the two numbers is \frac{1}{2}*-100 = -50. 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>

$$r = -50 - u s = -50 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 2500

$$(-50 - u) (-50 + u) = 2500$$

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

$$2500 - u^2 = 2500$$

Simplify the expression by subtracting 2500 on both sides

$$-u^2 = 2500-2500 = 0$$

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

$$u^2 = 0 u = 0$$

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

$$r = s = -50$$