Solve d^2+14d+13

$$d ^ { 2 } + 14 d + 13$$

Factor

$$\left(d+1\right)\left(d+13\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as d^{2}+ad+bd+13. To find a and b, set up a system to be solved.

$$a+b=14 ab=1\times 13=13$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=13$$

Rewrite d^{2}+14d+13 as \left(d^{2}+d\right)+\left(13d+13\right).

$$\left(d^{2}+d\right)+\left(13d+13\right)$$

Factor out d in the first and 13 in the second group.

$$d\left(d+1\right)+13\left(d+1\right)$$

Factor out common term d+1 by using distributive property.

$$\left(d+1\right)\left(d+13\right)$$

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.

$$d^{2}+14d+13=0$$

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.

$$d=\frac{-14±\sqrt{14^{2}-4\times 13}}{2}$$

Square 14.

$$d=\frac{-14±\sqrt{196-4\times 13}}{2}$$

Multiply -4 times 13.

$$d=\frac{-14±\sqrt{196-52}}{2}$$

Add 196 to -52.

$$d=\frac{-14±\sqrt{144}}{2}$$

Take the square root of 144.

$$d=\frac{-14±12}{2}$$

Now solve the equation d=\frac{-14±12}{2} when ± is plus. Add -14 to 12.

$$d=-\frac{2}{2}$$

Divide -2 by 2.

$$d=-1$$

Now solve the equation d=\frac{-14±12}{2} when ± is minus. Subtract 12 from -14.

$$d=-\frac{26}{2}$$

Divide -26 by 2.

$$d=-13$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and -13 for x_{2}.

$$d^{2}+14d+13=\left(d-\left(-1\right)\right)\left(d-\left(-13\right)\right)$$

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

$$d^{2}+14d+13=\left(d+1\right)\left(d+13\right)$$

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.

$$x ^ 2 +14x +13 = 0$$

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

$$r + s = -14 rs = 13$$

Two numbers r and s sum up to -14 exactly when the average of the two numbers is \frac{1}{2}*-14 = -7. 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>

$$r = -7 - u s = -7 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 13

$$(-7 - u) (-7 + u) = 13$$

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

$$49 - u^2 = 13$$

Simplify the expression by subtracting 49 on both sides

$$-u^2 = 13-49 = -36$$

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

$$u^2 = 36 u = \pm\sqrt{36} = \pm 6$$

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

$$r =-7 - 6 = -13 s = -7 + 6 = -1$$

Evaluate

$$\left(d+1\right)\left(d+13\right)$$