Solve a^2+2a-24

$$a ^ { 2 } + 2 a - 24$$

Factor

$$\left(a-4\right)\left(a+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-24. To find p and q, set up a system to be solved.

$$p+q=2 pq=1\left(-24\right)=-24$$

Since pq is negative, p and q have the opposite signs. Since p+q is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -24.

$$-1,24 -2,12 -3,8 -4,6$$

Calculate the sum for each pair.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

The solution is the pair that gives sum 2.

$$p=-4 q=6$$

Rewrite a^{2}+2a-24 as \left(a^{2}-4a\right)+\left(6a-24\right).

$$\left(a^{2}-4a\right)+\left(6a-24\right)$$

Factor out a in the first and 6 in the second group.

$$a\left(a-4\right)+6\left(a-4\right)$$

Factor out common term a-4 by using distributive property.

$$\left(a-4\right)\left(a+6\right)$$

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.

$$a^{2}+2a-24=0$$

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.

$$a=\frac{-2±\sqrt{2^{2}-4\left(-24\right)}}{2}$$

Square 2.

$$a=\frac{-2±\sqrt{4-4\left(-24\right)}}{2}$$

Multiply -4 times -24.

$$a=\frac{-2±\sqrt{4+96}}{2}$$

Add 4 to 96.

$$a=\frac{-2±\sqrt{100}}{2}$$

Take the square root of 100.

$$a=\frac{-2±10}{2}$$

Now solve the equation a=\frac{-2±10}{2} when ± is plus. Add -2 to 10.

$$a=\frac{8}{2}$$

Divide 8 by 2.

$$a=4$$

Now solve the equation a=\frac{-2±10}{2} when ± is minus. Subtract 10 from -2.

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

Divide -12 by 2.

$$a=-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4 for x_{1} and -6 for x_{2}.

$$a^{2}+2a-24=\left(a-4\right)\left(a-\left(-6\right)\right)$$

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

$$a^{2}+2a-24=\left(a-4\right)\left(a+6\right)$$

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.

$$x ^ 2 +2x -24 = 0$$

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

$$r + s = -2 rs = -24$$

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

$$r = -1 - u s = -1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -24

$$(-1 - u) (-1 + u) = -24$$

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

$$1 - u^2 = -24$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -24-1 = -25$$

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

$$u^2 = 25 u = \pm\sqrt{25} = \pm 5$$

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

$$r =-1 - 5 = -6 s = -1 + 5 = 4$$

Evaluate

$$\left(a-4\right)\left(a+6\right)$$