Solve x^2+19x+78

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

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.

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

Evaluate

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