Solve x^2-30x-2800

$$x ^ { 2 } - 30 x - 2800$$

Factor

$$\left(x-70\right)\left(x+40\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-2800. To find a and b, set up a system to be solved.

$$a+b=-30 ab=1\left(-2800\right)=-2800$$

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 -2800.

$$1,-2800 2,-1400 4,-700 5,-560 7,-400 8,-350 10,-280 14,-200 16,-175 20,-140 25,-112 28,-100 35,-80 40,-70 50,-56$$

Calculate the sum for each pair.

$$1-2800=-2799 2-1400=-1398 4-700=-696 5-560=-555 7-400=-393 8-350=-342 10-280=-270 14-200=-186 16-175=-159 20-140=-120 25-112=-87 28-100=-72 35-80=-45 40-70=-30 50-56=-6$$

The solution is the pair that gives sum -30.

$$a=-70 b=40$$

Rewrite x^{2}-30x-2800 as \left(x^{2}-70x\right)+\left(40x-2800\right).

$$\left(x^{2}-70x\right)+\left(40x-2800\right)$$

Factor out x in the first and 40 in the second group.

$$x\left(x-70\right)+40\left(x-70\right)$$

Factor out common term x-70 by using distributive property.

$$\left(x-70\right)\left(x+40\right)$$

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.

$$x^{2}-30x-2800=0$$

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.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\left(-2800\right)}}{2}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\left(-2800\right)}}{2}$$

Multiply -4 times -2800.

$$x=\frac{-\left(-30\right)±\sqrt{900+11200}}{2}$$

Add 900 to 11200.

$$x=\frac{-\left(-30\right)±\sqrt{12100}}{2}$$

Take the square root of 12100.

$$x=\frac{-\left(-30\right)±110}{2}$$

The opposite of -30 is 30.

$$x=\frac{30±110}{2}$$

Now solve the equation x=\frac{30±110}{2} when ± is plus. Add 30 to 110.

$$x=\frac{140}{2}$$

Divide 140 by 2.

$$x=70$$

Now solve the equation x=\frac{30±110}{2} when ± is minus. Subtract 110 from 30.

$$x=-\frac{80}{2}$$

Divide -80 by 2.

$$x=-40$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 70 for x_{1} and -40 for x_{2}.

$$x^{2}-30x-2800=\left(x-70\right)\left(x-\left(-40\right)\right)$$

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

$$x^{2}-30x-2800=\left(x-70\right)\left(x+40\right)$$

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.

$$x ^ 2 -30x -2800 = 0$$

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

$$r + s = 30 rs = -2800$$

Two numbers r and s sum up to 30 exactly when the average of the two numbers is \frac{1}{2}*30 = 15. 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>

$$r = 15 - u s = 15 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -2800

$$(15 - u) (15 + u) = -2800$$

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

$$225 - u^2 = -2800$$

Simplify the expression by subtracting 225 on both sides

$$-u^2 = -2800-225 = -3025$$

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

$$u^2 = 3025 u = \pm\sqrt{3025} = \pm 55$$

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

$$r =15 - 55 = -40 s = 15 + 55 = 70$$

Evaluate

$$\left(x-70\right)\left(x+40\right)$$