Solve n^2-4n-60

$$n ^ { 2 } - 4 n - 60$$

Factor

$$\left(n-10\right)\left(n+6\right)$$

Solution Steps

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

$$a+b=-4 ab=1\left(-60\right)=-60$$

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 -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 -4.

$$a=-10 b=6$$

Rewrite n^{2}-4n-60 as \left(n^{2}-10n\right)+\left(6n-60\right).

$$\left(n^{2}-10n\right)+\left(6n-60\right)$$

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

$$n\left(n-10\right)+6\left(n-10\right)$$

Factor out common term n-10 by using distributive property.

$$\left(n-10\right)\left(n+6\right)$$

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.

$$n^{2}-4n-60=0$$

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.

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

Square -4.

$$n=\frac{-\left(-4\right)±\sqrt{16-4\left(-60\right)}}{2}$$

Multiply -4 times -60.

$$n=\frac{-\left(-4\right)±\sqrt{16+240}}{2}$$

Add 16 to 240.

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

Take the square root of 256.

$$n=\frac{-\left(-4\right)±16}{2}$$

The opposite of -4 is 4.

$$n=\frac{4±16}{2}$$

Now solve the equation n=\frac{4±16}{2} when ± is plus. Add 4 to 16.

$$n=\frac{20}{2}$$

Divide 20 by 2.

$$n=10$$

Now solve the equation n=\frac{4±16}{2} when ± is minus. Subtract 16 from 4.

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

Divide -12 by 2.

$$n=-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 10 for x_{1} and -6 for x_{2}.

$$n^{2}-4n-60=\left(n-10\right)\left(n-\left(-6\right)\right)$$

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

$$n^{2}-4n-60=\left(n-10\right)\left(n+6\right)$$

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.

$$x ^ 2 -4x -60 = 0$$

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

$$r + s = 4 rs = -60$$

Two numbers r and s sum up to 4 exactly when the average of the two numbers is \frac{1}{2}*4 = 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>

$$r = 2 - u s = 2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -60

$$(2 - u) (2 + u) = -60$$

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

$$4 - u^2 = -60$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -60-4 = -64$$

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

$$u^2 = 64 u = \pm\sqrt{64} = \pm 8$$

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

$$r =2 - 8 = -6 s = 2 + 8 = 10$$

Evaluate

$$\left(n-10\right)\left(n+6\right)$$