Solve 3a^2+5a-2

$$3 { a }^{ 2 } +5a-2$$

Factor

$$\left(3a-1\right)\left(a+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3a^{2}+pa+qa-2. To find p and q, set up a system to be solved.

$$p+q=5 pq=3\left(-2\right)=-6$$

Since pq is negative, p and q have the opposite signs. Since p+q is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -6.

$$-1,6 -2,3$$

Calculate the sum for each pair.

$$-1+6=5 -2+3=1$$

The solution is the pair that gives sum 5.

$$p=-1 q=6$$

Rewrite 3a^{2}+5a-2 as \left(3a^{2}-a\right)+\left(6a-2\right).

$$\left(3a^{2}-a\right)+\left(6a-2\right)$$

Factor out a in the first and 2 in the second group.

$$a\left(3a-1\right)+2\left(3a-1\right)$$

Factor out common term 3a-1 by using distributive property.

$$\left(3a-1\right)\left(a+2\right)$$

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.

$$3a^{2}+5a-2=0$$

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.

$$a=\frac{-5±\sqrt{5^{2}-4\times 3\left(-2\right)}}{2\times 3}$$

Square 5.

$$a=\frac{-5±\sqrt{25-4\times 3\left(-2\right)}}{2\times 3}$$

Multiply -4 times 3.

$$a=\frac{-5±\sqrt{25-12\left(-2\right)}}{2\times 3}$$

Multiply -12 times -2.

$$a=\frac{-5±\sqrt{25+24}}{2\times 3}$$

Add 25 to 24.

$$a=\frac{-5±\sqrt{49}}{2\times 3}$$

Take the square root of 49.

$$a=\frac{-5±7}{2\times 3}$$

Multiply 2 times 3.

$$a=\frac{-5±7}{6}$$

Now solve the equation a=\frac{-5±7}{6} when ± is plus. Add -5 to 7.

$$a=\frac{2}{6}$$

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

$$a=\frac{1}{3}$$

Now solve the equation a=\frac{-5±7}{6} when ± is minus. Subtract 7 from -5.

$$a=-\frac{12}{6}$$

Divide -12 by 6.

$$a=-2$$

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

$$3a^{2}+5a-2=3\left(a-\frac{1}{3}\right)\left(a-\left(-2\right)\right)$$

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

$$3a^{2}+5a-2=3\left(a-\frac{1}{3}\right)\left(a+2\right)$$

Subtract \frac{1}{3} from a by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$3a^{2}+5a-2=3\times \frac{3a-1}{3}\left(a+2\right)$$

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

$$3a^{2}+5a-2=\left(3a-1\right)\left(a+2\right)$$

Evaluate

$$\left(3a-1\right)\left(a+2\right)$$