Solve 6x^2+3x+24

$$6 { x }^{ 2 } +3x+24$$

Factor

$$3\left(2x^{2}+x+8\right)$$

Solution Steps

Factor out 3. Polynomial 2x^{2}+x+8 is not factored since it does not have any rational roots.

$$3\left(2x^{2}+x+8\right)$$

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.

$$6x^{2}+3x+24=0$$

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.

$$x=\frac{-3±\sqrt{3^{2}-4\times 6\times 24}}{2\times 6}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 6\times 24}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-3±\sqrt{9-24\times 24}}{2\times 6}$$

Multiply -24 times 24.

$$x=\frac{-3±\sqrt{9-576}}{2\times 6}$$

Add 9 to -576.

$$x=\frac{-3±\sqrt{-567}}{2\times 6}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Quadratic polynomial cannot be factored.

$$6x^{2}+3x+24$$

Evaluate

$$3\left(2x^{2}+x+8\right)$$