Solve (3x^2-3x-53)+(9x^2+8x)=

$$(3 { x }^{ 2 } -3x-53)+(9 { x }^{ 2 } +8x)=$$

Evaluate

$$12x^{2}+5x-53$$

Solution Steps

Combine 3x^{2} and 9x^{2} to get 12x^{2}.

$$12x^{2}-3x-53+8x$$

Combine -3x and 8x to get 5x.

$$12x^{2}+5x-53$$

Factor

$$12\left(x-\frac{-\sqrt{2569}-5}{24}\right)\left(x-\frac{\sqrt{2569}-5}{24}\right)$$

Steps Using the Quadratic Formula

Combine 3x^{2} and 9x^{2} to get 12x^{2}.

$$factor(12x^{2}-3x-53+8x)$$

Combine -3x and 8x to get 5x.

$$factor(12x^{2}+5x-53)$$

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.

$$12x^{2}+5x-53=0$$

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.

$$x=\frac{-5±\sqrt{5^{2}-4\times 12\left(-53\right)}}{2\times 12}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 12\left(-53\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-5±\sqrt{25-48\left(-53\right)}}{2\times 12}$$

Multiply -48 times -53.

$$x=\frac{-5±\sqrt{25+2544}}{2\times 12}$$

Add 25 to 2544.

$$x=\frac{-5±\sqrt{2569}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{-5±\sqrt{2569}}{24}$$

Now solve the equation x=\frac{-5±\sqrt{2569}}{24} when ± is plus. Add -5 to \sqrt{2569}.

$$x=\frac{\sqrt{2569}-5}{24}$$

Now solve the equation x=\frac{-5±\sqrt{2569}}{24} when ± is minus. Subtract \sqrt{2569} from -5.

$$x=\frac{-\sqrt{2569}-5}{24}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5+\sqrt{2569}}{24} for x_{1} and \frac{-5-\sqrt{2569}}{24} for x_{2}.

$$12x^{2}+5x-53=12\left(x-\frac{\sqrt{2569}-5}{24}\right)\left(x-\frac{-\sqrt{2569}-5}{24}\right)$$