Solve 3x^2+26x-9

$$3 { x }^{ 2 } +26x-9$$

Factor

$$\left(3x-1\right)\left(x+9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx-9. To find a and b, set up a system to be solved.

$$a+b=26 ab=3\left(-9\right)=-27$$

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 -27.

$$-1,27 -3,9$$

Calculate the sum for each pair.

$$-1+27=26 -3+9=6$$

The solution is the pair that gives sum 26.

$$a=-1 b=27$$

Rewrite 3x^{2}+26x-9 as \left(3x^{2}-x\right)+\left(27x-9\right).

$$\left(3x^{2}-x\right)+\left(27x-9\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(3x-1\right)+9\left(3x-1\right)$$

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

$$\left(3x-1\right)\left(x+9\right)$$

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.

$$3x^{2}+26x-9=0$$

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.

$$x=\frac{-26±\sqrt{26^{2}-4\times 3\left(-9\right)}}{2\times 3}$$

Square 26.

$$x=\frac{-26±\sqrt{676-4\times 3\left(-9\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-26±\sqrt{676-12\left(-9\right)}}{2\times 3}$$

Multiply -12 times -9.

$$x=\frac{-26±\sqrt{676+108}}{2\times 3}$$

Add 676 to 108.

$$x=\frac{-26±\sqrt{784}}{2\times 3}$$

Take the square root of 784.

$$x=\frac{-26±28}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-26±28}{6}$$

Now solve the equation x=\frac{-26±28}{6} when ± is plus. Add -26 to 28.

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

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

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

Now solve the equation x=\frac{-26±28}{6} when ± is minus. Subtract 28 from -26.

$$x=-\frac{54}{6}$$

Divide -54 by 6.

$$x=-9$$

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 -9 for x_{2}.

$$3x^{2}+26x-9=3\left(x-\frac{1}{3}\right)\left(x-\left(-9\right)\right)$$

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

$$3x^{2}+26x-9=3\left(x-\frac{1}{3}\right)\left(x+9\right)$$

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

$$3x^{2}+26x-9=3\times \frac{3x-1}{3}\left(x+9\right)$$

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

$$3x^{2}+26x-9=\left(3x-1\right)\left(x+9\right)$$

Evaluate

$$\left(3x-1\right)\left(x+9\right)$$