Solve x^2+17x+16

$$x ^ { 2 } + 17 x + 16$$

Factor

$$\left(x+1\right)\left(x+16\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+16. To find a and b, set up a system to be solved.

$$a+b=17 ab=1\times 16=16$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 16.

$$1,16 2,8 4,4$$

Calculate the sum for each pair.

$$1+16=17 2+8=10 4+4=8$$

The solution is the pair that gives sum 17.

$$a=1 b=16$$

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

$$\left(x^{2}+x\right)+\left(16x+16\right)$$

Factor out x in the first and 16 in the second group.

$$x\left(x+1\right)+16\left(x+1\right)$$

Factor out common term x+1 by using distributive property.

$$\left(x+1\right)\left(x+16\right)$$

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.

$$x^{2}+17x+16=0$$

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.

$$x=\frac{-17±\sqrt{17^{2}-4\times 16}}{2}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\times 16}}{2}$$

Multiply -4 times 16.

$$x=\frac{-17±\sqrt{289-64}}{2}$$

Add 289 to -64.

$$x=\frac{-17±\sqrt{225}}{2}$$

Take the square root of 225.

$$x=\frac{-17±15}{2}$$

Now solve the equation x=\frac{-17±15}{2} when ± is plus. Add -17 to 15.

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

Divide -2 by 2.

$$x=-1$$

Now solve the equation x=\frac{-17±15}{2} when ± is minus. Subtract 15 from -17.

$$x=-\frac{32}{2}$$

Divide -32 by 2.

$$x=-16$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and -16 for x_{2}.

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

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

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

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.

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

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

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

Two numbers r and s sum up to -17 exactly when the average of the two numbers is \frac{1}{2}*-17 = -\frac{17}{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>

$$r = -\frac{17}{2} - u s = -\frac{17}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 16

$$(-\frac{17}{2} - u) (-\frac{17}{2} + u) = 16$$

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

$$\frac{289}{4} - u^2 = 16$$

Simplify the expression by subtracting \frac{289}{4} on both sides

$$-u^2 = 16-\frac{289}{4} = -\frac{225}{4}$$

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

$$u^2 = \frac{225}{4} u = \pm\sqrt{\frac{225}{4}} = \pm \frac{15}{2}$$

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

$$r =-\frac{17}{2} - \frac{15}{2} = -16 s = -\frac{17}{2} + \frac{15}{2} = -1$$

Evaluate

$$\left(x+1\right)\left(x+16\right)$$