Solve 6k^2-24k-270

$$6 k ^ { 2 } - 24 k - 270$$

Factor

$$6\left(k-9\right)\left(k+5\right)$$

Solution Steps

Factor out 6.

$$6\left(k^{2}-4k-45\right)$$

Consider k^{2}-4k-45. Factor the expression by grouping. First, the expression needs to be rewritten as k^{2}+ak+bk-45. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\left(-45\right)=-45$$

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 -45.

$$1,-45 3,-15 5,-9$$

Calculate the sum for each pair.

$$1-45=-44 3-15=-12 5-9=-4$$

The solution is the pair that gives sum -4.

$$a=-9 b=5$$

Rewrite k^{2}-4k-45 as \left(k^{2}-9k\right)+\left(5k-45\right).

$$\left(k^{2}-9k\right)+\left(5k-45\right)$$

Factor out k in the first and 5 in the second group.

$$k\left(k-9\right)+5\left(k-9\right)$$

Factor out common term k-9 by using distributive property.

$$\left(k-9\right)\left(k+5\right)$$

Rewrite the complete factored expression.

$$6\left(k-9\right)\left(k+5\right)$$

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.

$$6k^{2}-24k-270=0$$

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.

$$k=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 6\left(-270\right)}}{2\times 6}$$

Square -24.

$$k=\frac{-\left(-24\right)±\sqrt{576-4\times 6\left(-270\right)}}{2\times 6}$$

Multiply -4 times 6.

$$k=\frac{-\left(-24\right)±\sqrt{576-24\left(-270\right)}}{2\times 6}$$

Multiply -24 times -270.

$$k=\frac{-\left(-24\right)±\sqrt{576+6480}}{2\times 6}$$

Add 576 to 6480.

$$k=\frac{-\left(-24\right)±\sqrt{7056}}{2\times 6}$$

Take the square root of 7056.

$$k=\frac{-\left(-24\right)±84}{2\times 6}$$

The opposite of -24 is 24.

$$k=\frac{24±84}{2\times 6}$$

Multiply 2 times 6.

$$k=\frac{24±84}{12}$$

Now solve the equation k=\frac{24±84}{12} when ± is plus. Add 24 to 84.

$$k=\frac{108}{12}$$

Divide 108 by 12.

$$k=9$$

Now solve the equation k=\frac{24±84}{12} when ± is minus. Subtract 84 from 24.

$$k=-\frac{60}{12}$$

Divide -60 by 12.

$$k=-5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 9 for x_{1} and -5 for x_{2}.

$$6k^{2}-24k-270=6\left(k-9\right)\left(k-\left(-5\right)\right)$$

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

$$6k^{2}-24k-270=6\left(k-9\right)\left(k+5\right)$$

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.This is achieved by dividing both sides of the equation by 6

$$x ^ 2 -4x -45 = 0$$

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

$$r + s = 4 rs = -45$$

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-gzdabgg4ehffg0hf.b01.azurefd.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 = -45

$$(2 - u) (2 + u) = -45$$

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

$$4 - u^2 = -45$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -45-4 = -49$$

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

$$u^2 = 49 u = \pm\sqrt{49} = \pm 7$$

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

$$r =2 - 7 = -5 s = 2 + 7 = 9$$

Evaluate

$$6\left(k-9\right)\left(k+5\right)$$