z^2-7z+6-এর সমাধান করুন

$$z ^ { 2 } - 7 z + 6$$

Factor

$$\left(z-6\right)\left(z-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as z^{2}+az+bz+6. To find a and b, set up a system to be solved.

$$a+b=-7 ab=1\times 6=6$$

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 6.

$$-1,-6 -2,-3$$

Calculate the sum for each pair.

$$-1-6=-7 -2-3=-5$$

The solution is the pair that gives sum -7.

$$a=-6 b=-1$$

Rewrite z^{2}-7z+6 as \left(z^{2}-6z\right)+\left(-z+6\right).

$$\left(z^{2}-6z\right)+\left(-z+6\right)$$

Factor out z in the first and -1 in the second group.

$$z\left(z-6\right)-\left(z-6\right)$$

Factor out common term z-6 by using distributive property.

$$\left(z-6\right)\left(z-1\right)$$

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.

$$z^{2}-7z+6=0$$

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.

$$z=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6}}{2}$$

Square -7.

$$z=\frac{-\left(-7\right)±\sqrt{49-4\times 6}}{2}$$

Multiply -4 times 6.

$$z=\frac{-\left(-7\right)±\sqrt{49-24}}{2}$$

Add 49 to -24.

$$z=\frac{-\left(-7\right)±\sqrt{25}}{2}$$

Take the square root of 25.

$$z=\frac{-\left(-7\right)±5}{2}$$

The opposite of -7 is 7.

$$z=\frac{7±5}{2}$$

Now solve the equation z=\frac{7±5}{2} when ± is plus. Add 7 to 5.

$$z=\frac{12}{2}$$

Divide 12 by 2.

$$z=6$$

Now solve the equation z=\frac{7±5}{2} when ± is minus. Subtract 5 from 7.

$$z=\frac{2}{2}$$

Divide 2 by 2.

$$z=1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6 for x_{1} and 1 for x_{2}.

$$z^{2}-7z+6=\left(z-6\right)\left(z-1\right)$$

Evaluate

$$\left(z-6\right)\left(z-1\right)$$