Solve 100-20z+z^2

$$100 - 20 z + z ^ { 2 }$$

Factor

$$\left(z-10\right)^{2}$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$z^{2}-20z+100$$

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

$$a+b=-20 ab=1\times 100=100$$

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 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 -20.

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

Rewrite z^{2}-20z+100 as \left(z^{2}-10z\right)+\left(-10z+100\right).

$$\left(z^{2}-10z\right)+\left(-10z+100\right)$$

Factor out z in the first and -10 in the second group.

$$z\left(z-10\right)-10\left(z-10\right)$$

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

$$\left(z-10\right)\left(z-10\right)$$

Rewrite as a binomial square.

$$\left(z-10\right)^{2}$$

Steps Using Square Of Binomial

This trinomial has the form of a trinomial square, perhaps multiplied by a common factor. Trinomial squares can be factored by finding the square roots of the leading and trailing terms.

$$factor(z^{2}-20z+100)$$

Find the square root of the trailing term, 100.

$$\sqrt{100}=10$$

The trinomial square is the square of the binomial that is the sum or difference of the square roots of the leading and trailing terms, with the sign determined by the sign of the middle term of the trinomial square.

$$\left(z-10\right)^{2}$$

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.

$$z^{2}-20z+100=0$$

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.

$$z=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 100}}{2}$$

Square -20.

$$z=\frac{-\left(-20\right)±\sqrt{400-4\times 100}}{2}$$

Multiply -4 times 100.

$$z=\frac{-\left(-20\right)±\sqrt{400-400}}{2}$$

Add 400 to -400.

$$z=\frac{-\left(-20\right)±\sqrt{0}}{2}$$

Take the square root of 0.

$$z=\frac{-\left(-20\right)±0}{2}$$

The opposite of -20 is 20.

$$z=\frac{20±0}{2}$$

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 10 for x_{2}.

$$z^{2}-20z+100=\left(z-10\right)\left(z-10\right)$$

Evaluate

$$\left(z-10\right)^{2}$$