Solve 281x^2+512x-2448=0

$$281 x ^ { 2 } + 512 x - 2448 = 0$$

Solve for x

$$x=-4$$

$$x = \frac{612}{281} = 2\frac{50}{281} \approx 2.177935943$$

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 281x^{2}+ax+bx-2448. To find a and b, set up a system to be solved.

$$a+b=512 ab=281\left(-2448\right)=-687888$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -687888.

$$-1,687888 -2,343944 -3,229296 -4,171972 -6,114648 -8,85986 -9,76432 -12,57324 -16,42993 -17,40464 -18,38216 -24,28662 -34,20232 -36,19108 -48,14331 -51,13488 -68,10116 -72,9554 -102,6744 -136,5058 -144,4777 -153,4496 -204,3372 -272,2529 -281,2448 -306,2248 -408,1686 -562,1224 -612,1124 -816,843$$

Calculate the sum for each pair.

$$-1+687888=687887 -2+343944=343942 -3+229296=229293 -4+171972=171968 -6+114648=114642 -8+85986=85978 -9+76432=76423 -12+57324=57312 -16+42993=42977 -17+40464=40447 -18+38216=38198 -24+28662=28638 -34+20232=20198 -36+19108=19072 -48+14331=14283 -51+13488=13437 -68+10116=10048 -72+9554=9482 -102+6744=6642 -136+5058=4922 -144+4777=4633 -153+4496=4343 -204+3372=3168 -272+2529=2257 -281+2448=2167 -306+2248=1942 -408+1686=1278 -562+1224=662 -612+1124=512 -816+843=27$$

The solution is the pair that gives sum 512.

$$a=-612 b=1124$$

Rewrite 281x^{2}+512x-2448 as \left(281x^{2}-612x\right)+\left(1124x-2448\right).

$$\left(281x^{2}-612x\right)+\left(1124x-2448\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(281x-612\right)+4\left(281x-612\right)$$

Factor out common term 281x-612 by using distributive property.

$$\left(281x-612\right)\left(x+4\right)$$

To find equation solutions, solve 281x-612=0 and x+4=0.

$$x=\frac{612}{281} x=-4$$

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.

$$281x^{2}+512x-2448=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 281 for a, 512 for b, and -2448 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-512±\sqrt{512^{2}-4\times 281\left(-2448\right)}}{2\times 281}$$

Square 512.

$$x=\frac{-512±\sqrt{262144-4\times 281\left(-2448\right)}}{2\times 281}$$

Multiply -4 times 281.

$$x=\frac{-512±\sqrt{262144-1124\left(-2448\right)}}{2\times 281}$$

Multiply -1124 times -2448.

$$x=\frac{-512±\sqrt{262144+2751552}}{2\times 281}$$

Add 262144 to 2751552.

$$x=\frac{-512±\sqrt{3013696}}{2\times 281}$$

Take the square root of 3013696.

$$x=\frac{-512±1736}{2\times 281}$$

Multiply 2 times 281.

$$x=\frac{-512±1736}{562}$$

Now solve the equation x=\frac{-512±1736}{562} when ± is plus. Add -512 to 1736.

$$x=\frac{1224}{562}$$

Reduce the fraction \frac{1224}{562} to lowest terms by extracting and canceling out 2.

$$x=\frac{612}{281}$$

Now solve the equation x=\frac{-512±1736}{562} when ± is minus. Subtract 1736 from -512.

$$x=-\frac{2248}{562}$$

Divide -2248 by 562.

$$x=-4$$

The equation is now solved.

$$x=\frac{612}{281} x=-4$$

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.

$$281x^{2}+512x-2448=0$$

Add 2448 to both sides of the equation.

$$281x^{2}+512x-2448-\left(-2448\right)=-\left(-2448\right)$$

Subtracting -2448 from itself leaves 0.

$$281x^{2}+512x=-\left(-2448\right)$$

Subtract -2448 from 0.

$$281x^{2}+512x=2448$$

Divide both sides by 281.

$$\frac{281x^{2}+512x}{281}=\frac{2448}{281}$$

Dividing by 281 undoes the multiplication by 281.

$$x^{2}+\frac{512}{281}x=\frac{2448}{281}$$

Divide \frac{512}{281}, the coefficient of the x term, by 2 to get \frac{256}{281}. Then add the square of \frac{256}{281} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{512}{281}x+\left(\frac{256}{281}\right)^{2}=\frac{2448}{281}+\left(\frac{256}{281}\right)^{2}$$

Square \frac{256}{281} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{512}{281}x+\frac{65536}{78961}=\frac{2448}{281}+\frac{65536}{78961}$$

Add \frac{2448}{281} to \frac{65536}{78961} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{512}{281}x+\frac{65536}{78961}=\frac{753424}{78961}$$

Factor x^{2}+\frac{512}{281}x+\frac{65536}{78961}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{256}{281}\right)^{2}=\frac{753424}{78961}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{256}{281}\right)^{2}}=\sqrt{\frac{753424}{78961}}$$

Simplify.

$$x+\frac{256}{281}=\frac{868}{281} x+\frac{256}{281}=-\frac{868}{281}$$

Subtract \frac{256}{281} from both sides of the equation.

$$x=\frac{612}{281} x=-4$$

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 281

$$x ^ 2 +\frac{512}{281}x -\frac{2448}{281} = 0$$

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

$$r + s = -\frac{512}{281} rs = -\frac{2448}{281}$$

Two numbers r and s sum up to -\frac{512}{281} exactly when the average of the two numbers is \frac{1}{2}*-\frac{512}{281} = -\frac{256}{281}. 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>

$$r = -\frac{256}{281} - u s = -\frac{256}{281} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{2448}{281}

$$(-\frac{256}{281} - u) (-\frac{256}{281} + u) = -\frac{2448}{281}$$

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

$$\frac{65536}{78961} - u^2 = -\frac{2448}{281}$$

Simplify the expression by subtracting \frac{65536}{78961} on both sides

$$-u^2 = -\frac{2448}{281}-\frac{65536}{78961} = \frac{753424}{78961}$$

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

$$u^2 = -\frac{753424}{78961} u = \pm\sqrt{-\frac{753424}{78961}} = \pm \frac{868}{281}i$$

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

$$r =-\frac{256}{281} - \frac{868}{281}i = -4 s = -\frac{256}{281} + \frac{868}{281}i = 2.178$$