Solve a^2+33a+252

$$a ^ { 2 } + 33 a + 252$$

Factor

$$\left(a+12\right)\left(a+21\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa+252. To find p and q, set up a system to be solved.

$$p+q=33 pq=1\times 252=252$$

Since pq is positive, p and q have the same sign. Since p+q is positive, p and q are both positive. List all such integer pairs that give product 252.

$$1,252 2,126 3,84 4,63 6,42 7,36 9,28 12,21 14,18$$

Calculate the sum for each pair.

$$1+252=253 2+126=128 3+84=87 4+63=67 6+42=48 7+36=43 9+28=37 12+21=33 14+18=32$$

The solution is the pair that gives sum 33.

$$p=12 q=21$$

Rewrite a^{2}+33a+252 as \left(a^{2}+12a\right)+\left(21a+252\right).

$$\left(a^{2}+12a\right)+\left(21a+252\right)$$

Factor out a in the first and 21 in the second group.

$$a\left(a+12\right)+21\left(a+12\right)$$

Factor out common term a+12 by using distributive property.

$$\left(a+12\right)\left(a+21\right)$$

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.

$$a^{2}+33a+252=0$$

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.

$$a=\frac{-33±\sqrt{33^{2}-4\times 252}}{2}$$

Square 33.

$$a=\frac{-33±\sqrt{1089-4\times 252}}{2}$$

Multiply -4 times 252.

$$a=\frac{-33±\sqrt{1089-1008}}{2}$$

Add 1089 to -1008.

$$a=\frac{-33±\sqrt{81}}{2}$$

Take the square root of 81.

$$a=\frac{-33±9}{2}$$

Now solve the equation a=\frac{-33±9}{2} when ± is plus. Add -33 to 9.

$$a=-\frac{24}{2}$$

Divide -24 by 2.

$$a=-12$$

Now solve the equation a=\frac{-33±9}{2} when ± is minus. Subtract 9 from -33.

$$a=-\frac{42}{2}$$

Divide -42 by 2.

$$a=-21$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -12 for x_{1} and -21 for x_{2}.

$$a^{2}+33a+252=\left(a-\left(-12\right)\right)\left(a-\left(-21\right)\right)$$

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

$$a^{2}+33a+252=\left(a+12\right)\left(a+21\right)$$

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.

$$x ^ 2 +33x +252 = 0$$

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

$$r + s = -33 rs = 252$$

Two numbers r and s sum up to -33 exactly when the average of the two numbers is \frac{1}{2}*-33 = -\frac{33}{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>

$$r = -\frac{33}{2} - u s = -\frac{33}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 252

$$(-\frac{33}{2} - u) (-\frac{33}{2} + u) = 252$$

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

$$\frac{1089}{4} - u^2 = 252$$

Simplify the expression by subtracting \frac{1089}{4} on both sides

$$-u^2 = 252-\frac{1089}{4} = -\frac{81}{4}$$

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

$$u^2 = \frac{81}{4} u = \pm\sqrt{\frac{81}{4}} = \pm \frac{9}{2}$$

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

$$r =-\frac{33}{2} - \frac{9}{2} = -21 s = -\frac{33}{2} + \frac{9}{2} = -12$$

Evaluate

$$\left(a+12\right)\left(a+21\right)$$