Solve -3x^2+48x-210

$$-3 { x }^{ 2 } +48x-210$$

Factor

$$3\left(-x^{2}+16x-70\right)$$

Solution Steps

Factor out 3. Polynomial -x^{2}+16x-70 is not factored since it does not have any rational roots.

$$3\left(-x^{2}+16x-70\right)$$

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.

$$-3x^{2}+48x-210=0$$

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.

$$x=\frac{-48±\sqrt{48^{2}-4\left(-3\right)\left(-210\right)}}{2\left(-3\right)}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\left(-3\right)\left(-210\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-48±\sqrt{2304+12\left(-210\right)}}{2\left(-3\right)}$$

Multiply 12 times -210.

$$x=\frac{-48±\sqrt{2304-2520}}{2\left(-3\right)}$$

Add 2304 to -2520.

$$x=\frac{-48±\sqrt{-216}}{2\left(-3\right)}$$

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

$$-3x^{2}+48x-210$$

Evaluate

$$-3x^{2}+48x-210$$