Solve z^2-7z+6

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

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}.

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

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.

$$x ^ 2 -7x +6 = 0$$

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

$$r + s = 7 rs = 6$$

Two numbers r and s sum up to 7 exactly when the average of the two numbers is \frac{1}{2}*7 = \frac{7}{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>

$$r = \frac{7}{2} - u s = \frac{7}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 6

$$(\frac{7}{2} - u) (\frac{7}{2} + u) = 6$$

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

$$\frac{49}{4} - u^2 = 6$$

Simplify the expression by subtracting \frac{49}{4} on both sides

$$-u^2 = 6-\frac{49}{4} = -\frac{25}{4}$$

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

$$u^2 = \frac{25}{4} u = \pm\sqrt{\frac{25}{4}} = \pm \frac{5}{2}$$

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

$$r =\frac{7}{2} - \frac{5}{2} = 1 s = \frac{7}{2} + \frac{5}{2} = 6$$

Evaluate

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