Solve 100r^2+1200r+3500

$$100 r ^ { 2 } + 1200 r + 3500$$

Factor

$$100\left(r+5\right)\left(r+7\right)$$

Solution Steps

Factor out 100.

$$100\left(r^{2}+12r+35\right)$$

Consider r^{2}+12r+35. Factor the expression by grouping. First, the expression needs to be rewritten as r^{2}+ar+br+35. To find a and b, set up a system to be solved.

$$a+b=12 ab=1\times 35=35$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 35.

$$1,35 5,7$$

Calculate the sum for each pair.

$$1+35=36 5+7=12$$

The solution is the pair that gives sum 12.

$$a=5 b=7$$

Rewrite r^{2}+12r+35 as \left(r^{2}+5r\right)+\left(7r+35\right).

$$\left(r^{2}+5r\right)+\left(7r+35\right)$$

Factor out r in the first and 7 in the second group.

$$r\left(r+5\right)+7\left(r+5\right)$$

Factor out common term r+5 by using distributive property.

$$\left(r+5\right)\left(r+7\right)$$

Rewrite the complete factored expression.

$$100\left(r+5\right)\left(r+7\right)$$

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.

$$100r^{2}+1200r+3500=0$$

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.

$$r=\frac{-1200±\sqrt{1200^{2}-4\times 100\times 3500}}{2\times 100}$$

Square 1200.

$$r=\frac{-1200±\sqrt{1440000-4\times 100\times 3500}}{2\times 100}$$

Multiply -4 times 100.

$$r=\frac{-1200±\sqrt{1440000-400\times 3500}}{2\times 100}$$

Multiply -400 times 3500.

$$r=\frac{-1200±\sqrt{1440000-1400000}}{2\times 100}$$

Add 1440000 to -1400000.

$$r=\frac{-1200±\sqrt{40000}}{2\times 100}$$

Take the square root of 40000.

$$r=\frac{-1200±200}{2\times 100}$$

Multiply 2 times 100.

$$r=\frac{-1200±200}{200}$$

Now solve the equation r=\frac{-1200±200}{200} when ± is plus. Add -1200 to 200.

$$r=-\frac{1000}{200}$$

Divide -1000 by 200.

$$r=-5$$

Now solve the equation r=\frac{-1200±200}{200} when ± is minus. Subtract 200 from -1200.

$$r=-\frac{1400}{200}$$

Divide -1400 by 200.

$$r=-7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -5 for x_{1} and -7 for x_{2}.

$$100r^{2}+1200r+3500=100\left(r-\left(-5\right)\right)\left(r-\left(-7\right)\right)$$

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

$$100r^{2}+1200r+3500=100\left(r+5\right)\left(r+7\right)$$

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.This is achieved by dividing both sides of the equation by 100

$$x ^ 2 +12x +35 = 0$$

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

$$r + s = -12 rs = 35$$

Two numbers r and s sum up to -12 exactly when the average of the two numbers is \frac{1}{2}*-12 = -6. 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>

$$r = -6 - u s = -6 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 35

$$(-6 - u) (-6 + u) = 35$$

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

$$36 - u^2 = 35$$

Simplify the expression by subtracting 36 on both sides

$$-u^2 = 35-36 = -1$$

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

$$u^2 = 1 u = \pm\sqrt{1} = \pm 1$$

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

$$r =-6 - 1 = -7 s = -6 + 1 = -5$$

Evaluate

$$100\left(r+5\right)\left(r+7\right)$$