Solve j^2-16j-17

$$j ^ { 2 } - 16 j - 17$$

Factor

$$\left(j-17\right)\left(j+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as j^{2}+aj+bj-17. To find a and b, set up a system to be solved.

$$a+b=-16 ab=1\left(-17\right)=-17$$

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. The only such pair is the system solution.

$$a=-17 b=1$$

Rewrite j^{2}-16j-17 as \left(j^{2}-17j\right)+\left(j-17\right).

$$\left(j^{2}-17j\right)+\left(j-17\right)$$

Factor out j in j^{2}-17j.

$$j\left(j-17\right)+j-17$$

Factor out common term j-17 by using distributive property.

$$\left(j-17\right)\left(j+1\right)$$

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.

$$j^{2}-16j-17=0$$

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.

$$j=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-17\right)}}{2}$$

Square -16.

$$j=\frac{-\left(-16\right)±\sqrt{256-4\left(-17\right)}}{2}$$

Multiply -4 times -17.

$$j=\frac{-\left(-16\right)±\sqrt{256+68}}{2}$$

Add 256 to 68.

$$j=\frac{-\left(-16\right)±\sqrt{324}}{2}$$

Take the square root of 324.

$$j=\frac{-\left(-16\right)±18}{2}$$

The opposite of -16 is 16.

$$j=\frac{16±18}{2}$$

Now solve the equation j=\frac{16±18}{2} when ± is plus. Add 16 to 18.

$$j=\frac{34}{2}$$

Divide 34 by 2.

$$j=17$$

Now solve the equation j=\frac{16±18}{2} when ± is minus. Subtract 18 from 16.

$$j=-\frac{2}{2}$$

Divide -2 by 2.

$$j=-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 17 for x_{1} and -1 for x_{2}.

$$j^{2}-16j-17=\left(j-17\right)\left(j-\left(-1\right)\right)$$

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

$$j^{2}-16j-17=\left(j-17\right)\left(j+1\right)$$

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.

$$x ^ 2 -16x -17 = 0$$

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

$$r + s = 16 rs = -17$$

Two numbers r and s sum up to 16 exactly when the average of the two numbers is \frac{1}{2}*16 = 8. 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>

$$r = 8 - u s = 8 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -17

$$(8 - u) (8 + u) = -17$$

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

$$64 - u^2 = -17$$

Simplify the expression by subtracting 64 on both sides

$$-u^2 = -17-64 = -81$$

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

$$u^2 = 81 u = \pm\sqrt{81} = \pm 9$$

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

$$r =8 - 9 = -1 s = 8 + 9 = 17$$

Evaluate

$$\left(j-17\right)\left(j+1\right)$$