Solve a^2-a-30

$$a ^ { 2 } - a - 30$$

Factor

$$\left(a-6\right)\left(a+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-30. To find p and q, set up a system to be solved.

$$p+q=-1 pq=1\left(-30\right)=-30$$

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 -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 -1.

$$p=-6 q=5$$

Rewrite a^{2}-a-30 as \left(a^{2}-6a\right)+\left(5a-30\right).

$$\left(a^{2}-6a\right)+\left(5a-30\right)$$

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

$$a\left(a-6\right)+5\left(a-6\right)$$

Factor out common term a-6 by using distributive property.

$$\left(a-6\right)\left(a+5\right)$$

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.

$$a^{2}-a-30=0$$

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.

$$a=\frac{-\left(-1\right)±\sqrt{1-4\left(-30\right)}}{2}$$

Multiply -4 times -30.

$$a=\frac{-\left(-1\right)±\sqrt{1+120}}{2}$$

Add 1 to 120.

$$a=\frac{-\left(-1\right)±\sqrt{121}}{2}$$

Take the square root of 121.

$$a=\frac{-\left(-1\right)±11}{2}$$

The opposite of -1 is 1.

$$a=\frac{1±11}{2}$$

Now solve the equation a=\frac{1±11}{2} when ± is plus. Add 1 to 11.

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

Divide 12 by 2.

$$a=6$$

Now solve the equation a=\frac{1±11}{2} when ± is minus. Subtract 11 from 1.

$$a=-\frac{10}{2}$$

Divide -10 by 2.

$$a=-5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6 for x_{1} and -5 for x_{2}.

$$a^{2}-a-30=\left(a-6\right)\left(a-\left(-5\right)\right)$$

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

$$a^{2}-a-30=\left(a-6\right)\left(a+5\right)$$

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.

$$x ^ 2 -1x -30 = 0$$

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

$$r + s = 1 rs = -30$$

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

$$r = \frac{1}{2} - u s = \frac{1}{2} + u$$

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

$$(\frac{1}{2} - u) (\frac{1}{2} + u) = -30$$

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

$$\frac{1}{4} - u^2 = -30$$

Simplify the expression by subtracting \frac{1}{4} on both sides

$$-u^2 = -30-\frac{1}{4} = -\frac{121}{4}$$

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

$$u^2 = \frac{121}{4} u = \pm\sqrt{\frac{121}{4}} = \pm \frac{11}{2}$$

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

$$r =\frac{1}{2} - \frac{11}{2} = -5 s = \frac{1}{2} + \frac{11}{2} = 6$$

Evaluate

$$\left(a-6\right)\left(a+5\right)$$