Solve x^2+17x+42

$$x ^ { 2 } + 17 x + 42$$

Factor

$$\left(x+3\right)\left(x+14\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+42. To find a and b, set up a system to be solved.

$$a+b=17 ab=1\times 42=42$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. 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 17.

$$a=3 b=14$$

Rewrite x^{2}+17x+42 as \left(x^{2}+3x\right)+\left(14x+42\right).

$$\left(x^{2}+3x\right)+\left(14x+42\right)$$

Factor out x in the first and 14 in the second group.

$$x\left(x+3\right)+14\left(x+3\right)$$

Factor out common term x+3 by using distributive property.

$$\left(x+3\right)\left(x+14\right)$$

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.

$$x^{2}+17x+42=0$$

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.

$$x=\frac{-17±\sqrt{17^{2}-4\times 42}}{2}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\times 42}}{2}$$

Multiply -4 times 42.

$$x=\frac{-17±\sqrt{289-168}}{2}$$

Add 289 to -168.

$$x=\frac{-17±\sqrt{121}}{2}$$

Take the square root of 121.

$$x=\frac{-17±11}{2}$$

Now solve the equation x=\frac{-17±11}{2} when ± is plus. Add -17 to 11.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

Now solve the equation x=\frac{-17±11}{2} when ± is minus. Subtract 11 from -17.

$$x=-\frac{28}{2}$$

Divide -28 by 2.

$$x=-14$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and -14 for x_{2}.

$$x^{2}+17x+42=\left(x-\left(-3\right)\right)\left(x-\left(-14\right)\right)$$

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

$$x^{2}+17x+42=\left(x+3\right)\left(x+14\right)$$

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.

$$x ^ 2 +17x +42 = 0$$

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

$$r + s = -17 rs = 42$$

Two numbers r and s sum up to -17 exactly when the average of the two numbers is \frac{1}{2}*-17 = -\frac{17}{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>

$$r = -\frac{17}{2} - u s = -\frac{17}{2} + u$$

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

$$(-\frac{17}{2} - u) (-\frac{17}{2} + u) = 42$$

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

$$\frac{289}{4} - u^2 = 42$$

Simplify the expression by subtracting \frac{289}{4} on both sides

$$-u^2 = 42-\frac{289}{4} = -\frac{121}{4}$$

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.

$$r =-\frac{17}{2} - \frac{11}{2} = -14 s = -\frac{17}{2} + \frac{11}{2} = -3$$

Evaluate

$$\left(x+3\right)\left(x+14\right)$$