Solve {l}{y^2-18y}{+80}

$$\left. \begin{array} { l } { y ^ { 2 } - 18 y } \\ { + 80 } \end{array} \right.$$

Factor

$$\left(y-10\right)\left(y-8\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by+80. To find a and b, set up a system to be solved.

$$a+b=-18 ab=1\times 80=80$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 80.

$$-1,-80 -2,-40 -4,-20 -5,-16 -8,-10$$

Calculate the sum for each pair.

$$-1-80=-81 -2-40=-42 -4-20=-24 -5-16=-21 -8-10=-18$$

The solution is the pair that gives sum -18.

$$a=-10 b=-8$$

Rewrite y^{2}-18y+80 as \left(y^{2}-10y\right)+\left(-8y+80\right).

$$\left(y^{2}-10y\right)+\left(-8y+80\right)$$

Factor out y in the first and -8 in the second group.

$$y\left(y-10\right)-8\left(y-10\right)$$

Factor out common term y-10 by using distributive property.

$$\left(y-10\right)\left(y-8\right)$$

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.

$$y^{2}-18y+80=0$$

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.

$$y=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 80}}{2}$$

Square -18.

$$y=\frac{-\left(-18\right)±\sqrt{324-4\times 80}}{2}$$

Multiply -4 times 80.

$$y=\frac{-\left(-18\right)±\sqrt{324-320}}{2}$$

Add 324 to -320.

$$y=\frac{-\left(-18\right)±\sqrt{4}}{2}$$

Take the square root of 4.

$$y=\frac{-\left(-18\right)±2}{2}$$

The opposite of -18 is 18.

$$y=\frac{18±2}{2}$$

Now solve the equation y=\frac{18±2}{2} when ± is plus. Add 18 to 2.

$$y=\frac{20}{2}$$

Divide 20 by 2.

$$y=10$$

Now solve the equation y=\frac{18±2}{2} when ± is minus. Subtract 2 from 18.

$$y=\frac{16}{2}$$

Divide 16 by 2.

$$y=8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 10 for x_{1} and 8 for x_{2}.

$$y^{2}-18y+80=\left(y-10\right)\left(y-8\right)$$

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.

$$x ^ 2 -18x +80 = 0$$

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

$$r + s = 18 rs = 80$$

Two numbers r and s sum up to 18 exactly when the average of the two numbers is \frac{1}{2}*18 = 9. 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>

$$r = 9 - u s = 9 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 80

$$(9 - u) (9 + u) = 80$$

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

$$81 - u^2 = 80$$

Simplify the expression by subtracting 81 on both sides

$$-u^2 = 80-81 = -1$$

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.

$$r =9 - 1 = 8 s = 9 + 1 = 10$$

Evaluate

$$\left(y-10\right)\left(y-8\right)$$