Solve a^2-23a+42

$$a ^ { 2 } - 23 a + 42$$

Factor

$$\left(a-21\right)\left(a-2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa+42. To find p and q, set up a system to be solved.

$$p+q=-23 pq=1\times 42=42$$

Since pq is positive, p and q have the same sign. Since p+q is negative, p and q are both negative. List all such integer pairs that give product 42.

$$-1,-42 -2,-21 -3,-14 -6,-7$$

Calculate the sum for each pair.

$$-1-42=-43 -2-21=-23 -3-14=-17 -6-7=-13$$

The solution is the pair that gives sum -23.

$$p=-21 q=-2$$

Rewrite a^{2}-23a+42 as \left(a^{2}-21a\right)+\left(-2a+42\right).

$$\left(a^{2}-21a\right)+\left(-2a+42\right)$$

Factor out a in the first and -2 in the second group.

$$a\left(a-21\right)-2\left(a-21\right)$$

Factor out common term a-21 by using distributive property.

$$\left(a-21\right)\left(a-2\right)$$

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.

$$a^{2}-23a+42=0$$

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.

$$a=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 42}}{2}$$

Square -23.

$$a=\frac{-\left(-23\right)±\sqrt{529-4\times 42}}{2}$$

Multiply -4 times 42.

$$a=\frac{-\left(-23\right)±\sqrt{529-168}}{2}$$

Add 529 to -168.

$$a=\frac{-\left(-23\right)±\sqrt{361}}{2}$$

Take the square root of 361.

$$a=\frac{-\left(-23\right)±19}{2}$$

The opposite of -23 is 23.

$$a=\frac{23±19}{2}$$

Now solve the equation a=\frac{23±19}{2} when ± is plus. Add 23 to 19.

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

Divide 42 by 2.

$$a=21$$

Now solve the equation a=\frac{23±19}{2} when ± is minus. Subtract 19 from 23.

$$a=\frac{4}{2}$$

Divide 4 by 2.

$$a=2$$

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

$$a^{2}-23a+42=\left(a-21\right)\left(a-2\right)$$

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.

$$x ^ 2 -23x +42 = 0$$

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

$$r + s = 23 rs = 42$$

Two numbers r and s sum up to 23 exactly when the average of the two numbers is \frac{1}{2}*23 = \frac{23}{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>

$$r = \frac{23}{2} - u s = \frac{23}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 42

$$(\frac{23}{2} - u) (\frac{23}{2} + u) = 42$$

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

$$\frac{529}{4} - u^2 = 42$$

Simplify the expression by subtracting \frac{529}{4} on both sides

$$-u^2 = 42-\frac{529}{4} = -\frac{361}{4}$$

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

$$u^2 = \frac{361}{4} u = \pm\sqrt{\frac{361}{4}} = \pm \frac{19}{2}$$

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

$$r =\frac{23}{2} - \frac{19}{2} = 2 s = \frac{23}{2} + \frac{19}{2} = 21$$

Evaluate

$$\left(a-21\right)\left(a-2\right)$$