Solve x^2-7x-228

$$x ^ { 2 } - 7 x - 228$$

Factor

$$\left(x-19\right)\left(x+12\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-228. To find a and b, set up a system to be solved.

$$a+b=-7 ab=1\left(-228\right)=-228$$

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 -228.

$$1,-228 2,-114 3,-76 4,-57 6,-38 12,-19$$

Calculate the sum for each pair.

$$1-228=-227 2-114=-112 3-76=-73 4-57=-53 6-38=-32 12-19=-7$$

The solution is the pair that gives sum -7.

$$a=-19 b=12$$

Rewrite x^{2}-7x-228 as \left(x^{2}-19x\right)+\left(12x-228\right).

$$\left(x^{2}-19x\right)+\left(12x-228\right)$$

Factor out x in the first and 12 in the second group.

$$x\left(x-19\right)+12\left(x-19\right)$$

Factor out common term x-19 by using distributive property.

$$\left(x-19\right)\left(x+12\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$x^{2}-7x-228=0$$

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.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-228\right)}}{2}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-228\right)}}{2}$$

Multiply -4 times -228.

$$x=\frac{-\left(-7\right)±\sqrt{49+912}}{2}$$

Add 49 to 912.

$$x=\frac{-\left(-7\right)±\sqrt{961}}{2}$$

Take the square root of 961.

$$x=\frac{-\left(-7\right)±31}{2}$$

The opposite of -7 is 7.

$$x=\frac{7±31}{2}$$

Now solve the equation x=\frac{7±31}{2} when ± is plus. Add 7 to 31.

$$x=\frac{38}{2}$$

Divide 38 by 2.

$$x=19$$

Now solve the equation x=\frac{7±31}{2} when ± is minus. Subtract 31 from 7.

$$x=-\frac{24}{2}$$

Divide -24 by 2.

$$x=-12$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 19 for x_{1} and -12 for x_{2}.

$$x^{2}-7x-228=\left(x-19\right)\left(x-\left(-12\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$x^{2}-7x-228=\left(x-19\right)\left(x+12\right)$$

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.

$$x ^ 2 -7x -228 = 0$$

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

$$r + s = 7 rs = -228$$

Two numbers r and s sum up to 7 exactly when the average of the two numbers is \frac{1}{2}*7 = \frac{7}{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>

$$r = \frac{7}{2} - u s = \frac{7}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -228

$$(\frac{7}{2} - u) (\frac{7}{2} + u) = -228$$

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

$$\frac{49}{4} - u^2 = -228$$

Simplify the expression by subtracting \frac{49}{4} on both sides

$$-u^2 = -228-\frac{49}{4} = -\frac{961}{4}$$

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

$$u^2 = \frac{961}{4} u = \pm\sqrt{\frac{961}{4}} = \pm \frac{31}{2}$$

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

$$r =\frac{7}{2} - \frac{31}{2} = -12 s = \frac{7}{2} + \frac{31}{2} = 19$$

Evaluate

$$\left(x-19\right)\left(x+12\right)$$