Solve f(x)=x^2-13x-30?

$$f ( x ) = x ^ { 2 } - 13 x - 30 ?$$

Factor

$$\left(x-15\right)\left(x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-30. To find a and b, set up a system to be solved.

$$a+b=-13 ab=1\left(-30\right)=-30$$

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 -30.

$$1,-30 2,-15 3,-10 5,-6$$

Calculate the sum for each pair.

$$1-30=-29 2-15=-13 3-10=-7 5-6=-1$$

The solution is the pair that gives sum -13.

$$a=-15 b=2$$

Rewrite x^{2}-13x-30 as \left(x^{2}-15x\right)+\left(2x-30\right).

$$\left(x^{2}-15x\right)+\left(2x-30\right)$$

Factor out x in the first and 2 in the second group.

$$x\left(x-15\right)+2\left(x-15\right)$$

Factor out common term x-15 by using distributive property.

$$\left(x-15\right)\left(x+2\right)$$

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.

$$x^{2}-13x-30=0$$

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.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\left(-30\right)}}{2}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\left(-30\right)}}{2}$$

Multiply -4 times -30.

$$x=\frac{-\left(-13\right)±\sqrt{169+120}}{2}$$

Add 169 to 120.

$$x=\frac{-\left(-13\right)±\sqrt{289}}{2}$$

Take the square root of 289.

$$x=\frac{-\left(-13\right)±17}{2}$$

The opposite of -13 is 13.

$$x=\frac{13±17}{2}$$

Now solve the equation x=\frac{13±17}{2} when ± is plus. Add 13 to 17.

$$x=\frac{30}{2}$$

Divide 30 by 2.

$$x=15$$

Now solve the equation x=\frac{13±17}{2} when ± is minus. Subtract 17 from 13.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 15 for x_{1} and -2 for x_{2}.

$$x^{2}-13x-30=\left(x-15\right)\left(x-\left(-2\right)\right)$$

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

$$x^{2}-13x-30=\left(x-15\right)\left(x+2\right)$$

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.

$$x ^ 2 -13x -30 = 0$$

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

$$r + s = 13 rs = -30$$

Two numbers r and s sum up to 13 exactly when the average of the two numbers is \frac{1}{2}*13 = \frac{13}{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>

$$r = \frac{13}{2} - u s = \frac{13}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -30

$$(\frac{13}{2} - u) (\frac{13}{2} + u) = -30$$

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

$$\frac{169}{4} - u^2 = -30$$

Simplify the expression by subtracting \frac{169}{4} on both sides

$$-u^2 = -30-\frac{169}{4} = -\frac{289}{4}$$

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

$$u^2 = \frac{289}{4} u = \pm\sqrt{\frac{289}{4}} = \pm \frac{17}{2}$$

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

$$r =\frac{13}{2} - \frac{17}{2} = -2 s = \frac{13}{2} + \frac{17}{2} = 15$$

Evaluate

$$\left(x-15\right)\left(x+2\right)$$