Solve -x^2-241x+8130=0

$$- x ^ { 2 } - 241 x + 8130 = 0$$

Solve for x

$$x=-271$$

$$x=30$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+8130. To find a and b, set up a system to be solved.

$$a+b=-241 ab=-8130=-8130$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -8130.

$$1,-8130 2,-4065 3,-2710 5,-1626 6,-1355 10,-813 15,-542 30,-271$$

Calculate the sum for each pair.

$$1-8130=-8129 2-4065=-4063 3-2710=-2707 5-1626=-1621 6-1355=-1349 10-813=-803 15-542=-527 30-271=-241$$

The solution is the pair that gives sum -241.

$$a=30 b=-271$$

Rewrite -x^{2}-241x+8130 as \left(-x^{2}+30x\right)+\left(-271x+8130\right).

$$\left(-x^{2}+30x\right)+\left(-271x+8130\right)$$

Factor out x in the first and 271 in the second group.

$$x\left(-x+30\right)+271\left(-x+30\right)$$

Factor out common term -x+30 by using distributive property.

$$\left(-x+30\right)\left(x+271\right)$$

To find equation solutions, solve -x+30=0 and x+271=0.

$$x=30 x=-271$$

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.

$$-x^{2}-241x+8130=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -241 for b, and 8130 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-241\right)±\sqrt{\left(-241\right)^{2}-4\left(-1\right)\times 8130}}{2\left(-1\right)}$$

Square -241.

$$x=\frac{-\left(-241\right)±\sqrt{58081-4\left(-1\right)\times 8130}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-241\right)±\sqrt{58081+4\times 8130}}{2\left(-1\right)}$$

Multiply 4 times 8130.

$$x=\frac{-\left(-241\right)±\sqrt{58081+32520}}{2\left(-1\right)}$$

Add 58081 to 32520.

$$x=\frac{-\left(-241\right)±\sqrt{90601}}{2\left(-1\right)}$$

Take the square root of 90601.

$$x=\frac{-\left(-241\right)±301}{2\left(-1\right)}$$

The opposite of -241 is 241.

$$x=\frac{241±301}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{241±301}{-2}$$

Now solve the equation x=\frac{241±301}{-2} when ± is plus. Add 241 to 301.

$$x=\frac{542}{-2}$$

Divide 542 by -2.

$$x=-271$$

Now solve the equation x=\frac{241±301}{-2} when ± is minus. Subtract 301 from 241.

$$x=-\frac{60}{-2}$$

Divide -60 by -2.

$$x=30$$

The equation is now solved.

$$x=-271 x=30$$

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.

$$-x^{2}-241x+8130=0$$

Subtract 8130 from both sides of the equation.

$$-x^{2}-241x+8130-8130=-8130$$

Subtracting 8130 from itself leaves 0.

$$-x^{2}-241x=-8130$$

Divide both sides by -1.

$$\frac{-x^{2}-241x}{-1}=-\frac{8130}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{241}{-1}\right)x=-\frac{8130}{-1}$$

Divide -241 by -1.

$$x^{2}+241x=-\frac{8130}{-1}$$

Divide -8130 by -1.

$$x^{2}+241x=8130$$

Divide 241, the coefficient of the x term, by 2 to get \frac{241}{2}. Then add the square of \frac{241}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+241x+\left(\frac{241}{2}\right)^{2}=8130+\left(\frac{241}{2}\right)^{2}$$

Square \frac{241}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+241x+\frac{58081}{4}=8130+\frac{58081}{4}$$

Add 8130 to \frac{58081}{4}.

$$x^{2}+241x+\frac{58081}{4}=\frac{90601}{4}$$

Factor x^{2}+241x+\frac{58081}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{241}{2}\right)^{2}=\frac{90601}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{241}{2}\right)^{2}}=\sqrt{\frac{90601}{4}}$$

Simplify.

$$x+\frac{241}{2}=\frac{301}{2} x+\frac{241}{2}=-\frac{301}{2}$$

Subtract \frac{241}{2} from both sides of the equation.

$$x=30 x=-271$$

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.

$$x ^ 2 +241x -8130 = 0$$

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

$$r + s = -241 rs = -8130$$

Two numbers r and s sum up to -241 exactly when the average of the two numbers is \frac{1}{2}*-241 = -\frac{241}{2}. 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>

$$r = -\frac{241}{2} - u s = -\frac{241}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -8130

$$(-\frac{241}{2} - u) (-\frac{241}{2} + u) = -8130$$

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

$$\frac{58081}{4} - u^2 = -8130$$

Simplify the expression by subtracting \frac{58081}{4} on both sides

$$-u^2 = -8130-\frac{58081}{4} = -\frac{90601}{4}$$

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

$$u^2 = \frac{90601}{4} u = \pm\sqrt{\frac{90601}{4}} = \pm \frac{301}{2}$$

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

$$r =-\frac{241}{2} - \frac{301}{2} = -271 s = -\frac{241}{2} + \frac{301}{2} = 30$$