Solve 3x^2+14x-69

$$3 { x }^{ 2 } +14x-69$$

Factor

$$\left(x-3\right)\left(3x+23\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx-69. To find a and b, set up a system to be solved.

$$a+b=14 ab=3\left(-69\right)=-207$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -207.

$$-1,207 -3,69 -9,23$$

Calculate the sum for each pair.

$$-1+207=206 -3+69=66 -9+23=14$$

The solution is the pair that gives sum 14.

$$a=-9 b=23$$

Rewrite 3x^{2}+14x-69 as \left(3x^{2}-9x\right)+\left(23x-69\right).

$$\left(3x^{2}-9x\right)+\left(23x-69\right)$$

Factor out 3x in the first and 23 in the second group.

$$3x\left(x-3\right)+23\left(x-3\right)$$

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

$$\left(x-3\right)\left(3x+23\right)$$

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.

$$3x^{2}+14x-69=0$$

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.

$$x=\frac{-14±\sqrt{14^{2}-4\times 3\left(-69\right)}}{2\times 3}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 3\left(-69\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-14±\sqrt{196-12\left(-69\right)}}{2\times 3}$$

Multiply -12 times -69.

$$x=\frac{-14±\sqrt{196+828}}{2\times 3}$$

Add 196 to 828.

$$x=\frac{-14±\sqrt{1024}}{2\times 3}$$

Take the square root of 1024.

$$x=\frac{-14±32}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-14±32}{6}$$

Now solve the equation x=\frac{-14±32}{6} when ± is plus. Add -14 to 32.

$$x=\frac{18}{6}$$

Divide 18 by 6.

$$x=3$$

Now solve the equation x=\frac{-14±32}{6} when ± is minus. Subtract 32 from -14.

$$x=-\frac{46}{6}$$

Reduce the fraction \frac{-46}{6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{23}{3}$$

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 -\frac{23}{3} for x_{2}.

$$3x^{2}+14x-69=3\left(x-3\right)\left(x-\left(-\frac{23}{3}\right)\right)$$

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

$$3x^{2}+14x-69=3\left(x-3\right)\left(x+\frac{23}{3}\right)$$

Add \frac{23}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$3x^{2}+14x-69=3\left(x-3\right)\times \frac{3x+23}{3}$$

Cancel out 3, the greatest common factor in 3 and 3.

$$3x^{2}+14x-69=\left(x-3\right)\left(3x+23\right)$$

Evaluate

$$\left(x-3\right)\left(3x+23\right)$$