Solve (t^2+3)+(5t^2-4t-7)

$$( t ^ { 2 } + 3 ) + ( 5 t ^ { 2 } - 4 t - 7 )$$

Evaluate

$$6t^{2}-4t-4$$

Solution Steps

Combine t^{2} and 5t^{2} to get 6t^{2}.

$$6t^{2}+3-4t-7$$

Subtract 7 from 3 to get -4.

$$6t^{2}-4-4t$$

Factor

$$6\left(t-\frac{1-\sqrt{7}}{3}\right)\left(t-\frac{\sqrt{7}+1}{3}\right)$$

Steps Using the Quadratic Formula

Combine t^{2} and 5t^{2} to get 6t^{2}.

$$factor(6t^{2}+3-4t-7)$$

Subtract 7 from 3 to get -4.

$$factor(6t^{2}-4-4t)$$

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.

$$6t^{2}-4t-4=0$$

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.

$$t=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 6\left(-4\right)}}{2\times 6}$$

Square -4.

$$t=\frac{-\left(-4\right)±\sqrt{16-4\times 6\left(-4\right)}}{2\times 6}$$

Multiply -4 times 6.

$$t=\frac{-\left(-4\right)±\sqrt{16-24\left(-4\right)}}{2\times 6}$$

Multiply -24 times -4.

$$t=\frac{-\left(-4\right)±\sqrt{16+96}}{2\times 6}$$

Add 16 to 96.

$$t=\frac{-\left(-4\right)±\sqrt{112}}{2\times 6}$$

Take the square root of 112.

$$t=\frac{-\left(-4\right)±4\sqrt{7}}{2\times 6}$$

The opposite of -4 is 4.

$$t=\frac{4±4\sqrt{7}}{2\times 6}$$

Multiply 2 times 6.

$$t=\frac{4±4\sqrt{7}}{12}$$

Now solve the equation t=\frac{4±4\sqrt{7}}{12} when ± is plus. Add 4 to 4\sqrt{7}.

$$t=\frac{4\sqrt{7}+4}{12}$$

Divide 4+4\sqrt{7} by 12.

$$t=\frac{\sqrt{7}+1}{3}$$

Now solve the equation t=\frac{4±4\sqrt{7}}{12} when ± is minus. Subtract 4\sqrt{7} from 4.

$$t=\frac{4-4\sqrt{7}}{12}$$

Divide 4-4\sqrt{7} by 12.

$$t=\frac{1-\sqrt{7}}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{7}}{3} for x_{1} and \frac{1-\sqrt{7}}{3} for x_{2}.

$$6t^{2}-4t-4=6\left(t-\frac{\sqrt{7}+1}{3}\right)\left(t-\frac{1-\sqrt{7}}{3}\right)$$