Solve x^2+19x+78

$$x ^ { 2 } + 19 x + 78$$

Factor

$$\left(x+6\right)\left(x+13\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+78. To find a and b, set up a system to be solved.

$$a+b=19 ab=1\times 78=78$$

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 78.

$$1,78 2,39 3,26 6,13$$

Calculate the sum for each pair.

$$1+78=79 2+39=41 3+26=29 6+13=19$$

The solution is the pair that gives sum 19.

$$a=6 b=13$$

Rewrite x^{2}+19x+78 as \left(x^{2}+6x\right)+\left(13x+78\right).

$$\left(x^{2}+6x\right)+\left(13x+78\right)$$

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

$$x\left(x+6\right)+13\left(x+6\right)$$

Factor out common term x+6 by using distributive property.

$$\left(x+6\right)\left(x+13\right)$$

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.

$$x^{2}+19x+78=0$$

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.

$$x=\frac{-19±\sqrt{19^{2}-4\times 78}}{2}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\times 78}}{2}$$

Multiply -4 times 78.

$$x=\frac{-19±\sqrt{361-312}}{2}$$

Add 361 to -312.

$$x=\frac{-19±\sqrt{49}}{2}$$

Take the square root of 49.

$$x=\frac{-19±7}{2}$$

Now solve the equation x=\frac{-19±7}{2} when ± is plus. Add -19 to 7.

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

Divide -12 by 2.

$$x=-6$$

Now solve the equation x=\frac{-19±7}{2} when ± is minus. Subtract 7 from -19.

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

Divide -26 by 2.

$$x=-13$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -6 for x_{1} and -13 for x_{2}.

$$x^{2}+19x+78=\left(x-\left(-6\right)\right)\left(x-\left(-13\right)\right)$$

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

$$x^{2}+19x+78=\left(x+6\right)\left(x+13\right)$$

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.

$$x ^ 2 +19x +78 = 0$$

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

$$r + s = -19 rs = 78$$

Two numbers r and s sum up to -19 exactly when the average of the two numbers is \frac{1}{2}*-19 = -\frac{19}{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>

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

To solve for unknown quantity u, substitute these in the product equation rs = 78

$$(-\frac{19}{2} - u) (-\frac{19}{2} + u) = 78$$

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

$$\frac{361}{4} - u^2 = 78$$

Simplify the expression by subtracting \frac{361}{4} on both sides

$$-u^2 = 78-\frac{361}{4} = -\frac{49}{4}$$

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

$$u^2 = \frac{49}{4} u = \pm\sqrt{\frac{49}{4}} = \pm \frac{7}{2}$$

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

$$r =-\frac{19}{2} - \frac{7}{2} = -13 s = -\frac{19}{2} + \frac{7}{2} = -6$$

Evaluate

$$\left(x+6\right)\left(x+13\right)$$