Solve b^2-17b-60

$$b ^ { 2 } - 17 b - 60$$

Factor

$$\left(b-20\right)\left(b+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as b^{2}+pb+qb-60. To find p and q, set up a system to be solved.

$$p+q=-17 pq=1\left(-60\right)=-60$$

Since pq is negative, p and q have the opposite signs. Since p+q is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Calculate the sum for each pair.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

The solution is the pair that gives sum -17.

$$p=-20 q=3$$

Rewrite b^{2}-17b-60 as \left(b^{2}-20b\right)+\left(3b-60\right).

$$\left(b^{2}-20b\right)+\left(3b-60\right)$$

Factor out b in the first and 3 in the second group.

$$b\left(b-20\right)+3\left(b-20\right)$$

Factor out common term b-20 by using distributive property.

$$\left(b-20\right)\left(b+3\right)$$

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.

$$b^{2}-17b-60=0$$

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.

$$b=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\left(-60\right)}}{2}$$

Square -17.

$$b=\frac{-\left(-17\right)±\sqrt{289-4\left(-60\right)}}{2}$$

Multiply -4 times -60.

$$b=\frac{-\left(-17\right)±\sqrt{289+240}}{2}$$

Add 289 to 240.

$$b=\frac{-\left(-17\right)±\sqrt{529}}{2}$$

Take the square root of 529.

$$b=\frac{-\left(-17\right)±23}{2}$$

The opposite of -17 is 17.

$$b=\frac{17±23}{2}$$

Now solve the equation b=\frac{17±23}{2} when ± is plus. Add 17 to 23.

$$b=\frac{40}{2}$$

Divide 40 by 2.

$$b=20$$

Now solve the equation b=\frac{17±23}{2} when ± is minus. Subtract 23 from 17.

$$b=-\frac{6}{2}$$

Divide -6 by 2.

$$b=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 20 for x_{1} and -3 for x_{2}.

$$b^{2}-17b-60=\left(b-20\right)\left(b-\left(-3\right)\right)$$

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

$$b^{2}-17b-60=\left(b-20\right)\left(b+3\right)$$

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.

$$x ^ 2 -17x -60 = 0$$

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

$$r + s = 17 rs = -60$$

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 = -60

$$(\frac{17}{2} - u) (\frac{17}{2} + u) = -60$$

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

$$\frac{289}{4} - u^2 = -60$$

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

$$-u^2 = -60-\frac{289}{4} = -\frac{529}{4}$$

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

$$u^2 = \frac{529}{4} u = \pm\sqrt{\frac{529}{4}} = \pm \frac{23}{2}$$

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

$$r =\frac{17}{2} - \frac{23}{2} = -3 s = \frac{17}{2} + \frac{23}{2} = 20$$

Evaluate

$$\left(b-20\right)\left(b+3\right)$$