Solve y^2+101y+100

$$y ^ { 2 } + 101 y + 100$$

Factor

$$\left(y+1\right)\left(y+100\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by+100. To find a and b, set up a system to be solved.

$$a+b=101 ab=1\times 100=100$$

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 100.

$$1,100 2,50 4,25 5,20 10,10$$

Calculate the sum for each pair.

$$1+100=101 2+50=52 4+25=29 5+20=25 10+10=20$$

The solution is the pair that gives sum 101.

$$a=1 b=100$$

Rewrite y^{2}+101y+100 as \left(y^{2}+y\right)+\left(100y+100\right).

$$\left(y^{2}+y\right)+\left(100y+100\right)$$

Factor out y in the first and 100 in the second group.

$$y\left(y+1\right)+100\left(y+1\right)$$

Factor out common term y+1 by using distributive property.

$$\left(y+1\right)\left(y+100\right)$$

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.

$$y^{2}+101y+100=0$$

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.

$$y=\frac{-101±\sqrt{101^{2}-4\times 100}}{2}$$

Square 101.

$$y=\frac{-101±\sqrt{10201-4\times 100}}{2}$$

Multiply -4 times 100.

$$y=\frac{-101±\sqrt{10201-400}}{2}$$

Add 10201 to -400.

$$y=\frac{-101±\sqrt{9801}}{2}$$

Take the square root of 9801.

$$y=\frac{-101±99}{2}$$

Now solve the equation y=\frac{-101±99}{2} when ± is plus. Add -101 to 99.

$$y=-\frac{2}{2}$$

Divide -2 by 2.

$$y=-1$$

Now solve the equation y=\frac{-101±99}{2} when ± is minus. Subtract 99 from -101.

$$y=-\frac{200}{2}$$

Divide -200 by 2.

$$y=-100$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and -100 for x_{2}.

$$y^{2}+101y+100=\left(y-\left(-1\right)\right)\left(y-\left(-100\right)\right)$$

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

$$y^{2}+101y+100=\left(y+1\right)\left(y+100\right)$$

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.

$$x ^ 2 +101x +100 = 0$$

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

$$r + s = -101 rs = 100$$

Two numbers r and s sum up to -101 exactly when the average of the two numbers is \frac{1}{2}*-101 = -\frac{101}{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>

$$r = -\frac{101}{2} - u s = -\frac{101}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 100

$$(-\frac{101}{2} - u) (-\frac{101}{2} + u) = 100$$

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

$$\frac{10201}{4} - u^2 = 100$$

Simplify the expression by subtracting \frac{10201}{4} on both sides

$$-u^2 = 100-\frac{10201}{4} = -\frac{9801}{4}$$

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

$$u^2 = \frac{9801}{4} u = \pm\sqrt{\frac{9801}{4}} = \pm \frac{99}{2}$$

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

$$r =-\frac{101}{2} - \frac{99}{2} = -100 s = -\frac{101}{2} + \frac{99}{2} = -1$$

Evaluate

$$\left(y+1\right)\left(y+100\right)$$