Solve z^2-2z-48

$$z ^ { 2 } - 2 z - 48$$

Factor

$$\left(z-8\right)\left(z+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as z^{2}+az+bz-48. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-48\right)=-48$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Calculate the sum for each pair.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

The solution is the pair that gives sum -2.

$$a=-8 b=6$$

Rewrite z^{2}-2z-48 as \left(z^{2}-8z\right)+\left(6z-48\right).

$$\left(z^{2}-8z\right)+\left(6z-48\right)$$

Factor out z in the first and 6 in the second group.

$$z\left(z-8\right)+6\left(z-8\right)$$

Factor out common term z-8 by using distributive property.

$$\left(z-8\right)\left(z+6\right)$$

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.

$$z^{2}-2z-48=0$$

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.

$$z=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-48\right)}}{2}$$

Square -2.

$$z=\frac{-\left(-2\right)±\sqrt{4-4\left(-48\right)}}{2}$$

Multiply -4 times -48.

$$z=\frac{-\left(-2\right)±\sqrt{4+192}}{2}$$

Add 4 to 192.

$$z=\frac{-\left(-2\right)±\sqrt{196}}{2}$$

Take the square root of 196.

$$z=\frac{-\left(-2\right)±14}{2}$$

The opposite of -2 is 2.

$$z=\frac{2±14}{2}$$

Now solve the equation z=\frac{2±14}{2} when ± is plus. Add 2 to 14.

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

Divide 16 by 2.

$$z=8$$

Now solve the equation z=\frac{2±14}{2} when ± is minus. Subtract 14 from 2.

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

Divide -12 by 2.

$$z=-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8 for x_{1} and -6 for x_{2}.

$$z^{2}-2z-48=\left(z-8\right)\left(z-\left(-6\right)\right)$$

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

$$z^{2}-2z-48=\left(z-8\right)\left(z+6\right)$$

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.

$$x ^ 2 -2x -48 = 0$$

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

$$r + s = 2 rs = -48$$

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

$$r = 1 - u s = 1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -48

$$(1 - u) (1 + u) = -48$$

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

$$1 - u^2 = -48$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -48-1 = -49$$

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

$$u^2 = 49 u = \pm\sqrt{49} = \pm 7$$

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

$$r =1 - 7 = -6 s = 1 + 7 = 8$$

Evaluate

$$\left(z-8\right)\left(z+6\right)$$