Solve a^2-a-56

$$a ^ { 2 } - a - 56$$

Factor

$$\left(a-8\right)\left(a+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-56. To find p and q, set up a system to be solved.

$$p+q=-1 pq=1\left(-56\right)=-56$$

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 -56.

$$1,-56 2,-28 4,-14 7,-8$$

Calculate the sum for each pair.

$$1-56=-55 2-28=-26 4-14=-10 7-8=-1$$

The solution is the pair that gives sum -1.

$$p=-8 q=7$$

Rewrite a^{2}-a-56 as \left(a^{2}-8a\right)+\left(7a-56\right).

$$\left(a^{2}-8a\right)+\left(7a-56\right)$$

Factor out a in the first and 7 in the second group.

$$a\left(a-8\right)+7\left(a-8\right)$$

Factor out common term a-8 by using distributive property.

$$\left(a-8\right)\left(a+7\right)$$

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.

$$a^{2}-a-56=0$$

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.

$$a=\frac{-\left(-1\right)±\sqrt{1-4\left(-56\right)}}{2}$$

Multiply -4 times -56.

$$a=\frac{-\left(-1\right)±\sqrt{1+224}}{2}$$

Add 1 to 224.

$$a=\frac{-\left(-1\right)±\sqrt{225}}{2}$$

Take the square root of 225.

$$a=\frac{-\left(-1\right)±15}{2}$$

The opposite of -1 is 1.

$$a=\frac{1±15}{2}$$

Now solve the equation a=\frac{1±15}{2} when ± is plus. Add 1 to 15.

$$a=\frac{16}{2}$$

Divide 16 by 2.

$$a=8$$

Now solve the equation a=\frac{1±15}{2} when ± is minus. Subtract 15 from 1.

$$a=-\frac{14}{2}$$

Divide -14 by 2.

$$a=-7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8 for x_{1} and -7 for x_{2}.

$$a^{2}-a-56=\left(a-8\right)\left(a-\left(-7\right)\right)$$

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

$$a^{2}-a-56=\left(a-8\right)\left(a+7\right)$$

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.

$$x ^ 2 -1x -56 = 0$$

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

$$r + s = 1 rs = -56$$

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 = -56

$$(\frac{1}{2} - u) (\frac{1}{2} + u) = -56$$

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

$$\frac{1}{4} - u^2 = -56$$

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

$$-u^2 = -56-\frac{1}{4} = -\frac{225}{4}$$

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

$$u^2 = \frac{225}{4} u = \pm\sqrt{\frac{225}{4}} = \pm \frac{15}{2}$$

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

$$r =\frac{1}{2} - \frac{15}{2} = -7 s = \frac{1}{2} + \frac{15}{2} = 8$$

Evaluate

$$\left(a-8\right)\left(a+7\right)$$