Solve -4*9t^2+19*6t-9*6

$$- 4 \cdot 9 t ^ { 2 } + 19 \cdot 6 t - 9 \cdot 6$$

Evaluate

$$-36t^{2}+114t-54$$

Solution Steps

Do the multiplications.

$$-36t^{2}+114t-9\times 6$$

Multiply 9 and 6 to get 54.

$$-36t^{2}+114t-54$$

Factor

$$-36\left(t-\frac{19-\sqrt{145}}{12}\right)\left(t-\frac{\sqrt{145}+19}{12}\right)$$

Steps Using the Quadratic Formula

Do the multiplications.

$$factor(-36t^{2}+114t-9\times 6)$$

Multiply 9 and 6 to get 54.

$$factor(-36t^{2}+114t-54)$$

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.

$$-36t^{2}+114t-54=0$$

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.

$$t=\frac{-114±\sqrt{114^{2}-4\left(-36\right)\left(-54\right)}}{2\left(-36\right)}$$

Square 114.

$$t=\frac{-114±\sqrt{12996-4\left(-36\right)\left(-54\right)}}{2\left(-36\right)}$$

Multiply -4 times -36.

$$t=\frac{-114±\sqrt{12996+144\left(-54\right)}}{2\left(-36\right)}$$

Multiply 144 times -54.

$$t=\frac{-114±\sqrt{12996-7776}}{2\left(-36\right)}$$

Add 12996 to -7776.

$$t=\frac{-114±\sqrt{5220}}{2\left(-36\right)}$$

Take the square root of 5220.

$$t=\frac{-114±6\sqrt{145}}{2\left(-36\right)}$$

Multiply 2 times -36.

$$t=\frac{-114±6\sqrt{145}}{-72}$$

Now solve the equation t=\frac{-114±6\sqrt{145}}{-72} when ± is plus. Add -114 to 6\sqrt{145}.

$$t=\frac{6\sqrt{145}-114}{-72}$$

Divide -114+6\sqrt{145} by -72.

$$t=\frac{19-\sqrt{145}}{12}$$

Now solve the equation t=\frac{-114±6\sqrt{145}}{-72} when ± is minus. Subtract 6\sqrt{145} from -114.

$$t=\frac{-6\sqrt{145}-114}{-72}$$

Divide -114-6\sqrt{145} by -72.

$$t=\frac{\sqrt{145}+19}{12}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{19-\sqrt{145}}{12} for x_{1} and \frac{19+\sqrt{145}}{12} for x_{2}.

$$-36t^{2}+114t-54=-36\left(t-\frac{19-\sqrt{145}}{12}\right)\left(t-\frac{\sqrt{145}+19}{12}\right)$$