Solve k^2=16k-60

$$k ^ { 2 } = 16 k - 60$$

Solve for k

$$k=6$$

$$k=10$$

Steps Using Factoring

Subtract 16k from both sides.

$$k^{2}-16k=-60$$

Add 60 to both sides.

$$k^{2}-16k+60=0$$

To solve the equation, factor k^{2}-16k+60 using formula k^{2}+\left(a+b\right)k+ab=\left(k+a\right)\left(k+b\right). To find a and b, set up a system to be solved.

$$a+b=-16 ab=60$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. 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.

$$-1-60=-61 -2-30=-32 -3-20=-23 -4-15=-19 -5-12=-17 -6-10=-16$$

The solution is the pair that gives sum -16.

$$a=-10 b=-6$$

Rewrite factored expression \left(k+a\right)\left(k+b\right) using the obtained values.

$$\left(k-10\right)\left(k-6\right)$$

To find equation solutions, solve k-10=0 and k-6=0.

$$k=10 k=6$$

Steps Using Factoring By Grouping

Subtract 16k from both sides.

$$k^{2}-16k=-60$$

Add 60 to both sides.

$$k^{2}-16k+60=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as k^{2}+ak+bk+60. To find a and b, set up a system to be solved.

$$a+b=-16 ab=1\times 60=60$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. 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.

$$-1-60=-61 -2-30=-32 -3-20=-23 -4-15=-19 -5-12=-17 -6-10=-16$$

The solution is the pair that gives sum -16.

$$a=-10 b=-6$$

Rewrite k^{2}-16k+60 as \left(k^{2}-10k\right)+\left(-6k+60\right).

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

Factor out k in the first and -6 in the second group.

$$k\left(k-10\right)-6\left(k-10\right)$$

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

$$\left(k-10\right)\left(k-6\right)$$

To find equation solutions, solve k-10=0 and k-6=0.

$$k=10 k=6$$

Steps Using the Quadratic Formula

Subtract 16k from both sides.

$$k^{2}-16k=-60$$

Add 60 to both sides.

$$k^{2}-16k+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -16.

$$k=\frac{-\left(-16\right)±\sqrt{256-4\times 60}}{2}$$

Multiply -4 times 60.

$$k=\frac{-\left(-16\right)±\sqrt{256-240}}{2}$$

Add 256 to -240.

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

Take the square root of 16.

$$k=\frac{-\left(-16\right)±4}{2}$$

The opposite of -16 is 16.

$$k=\frac{16±4}{2}$$

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

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

Divide 20 by 2.

$$k=10$$

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

$$k=\frac{12}{2}$$

Divide 12 by 2.

$$k=6$$

The equation is now solved.

$$k=10 k=6$$

Steps for Completing the Square

Subtract 16k from both sides.

$$k^{2}-16k=-60$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}-16k+\left(-8\right)^{2}=-60+\left(-8\right)^{2}$$

Square -8.

$$k^{2}-16k+64=-60+64$$

Add -60 to 64.

$$k^{2}-16k+64=4$$

Factor k^{2}-16k+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(k-8\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k-8\right)^{2}}=\sqrt{4}$$

Simplify.

$$k-8=2 k-8=-2$$

Add 8 to both sides of the equation.

$$k=10 k=6$$