Solve 1.5x^2-52.5x-4500=0

$$1.5 x ^ { 2 } - 52.5 x - 4500 = 0$$

Solve for x

$$x=-40$$

$$x=75$$

Steps Using the Quadratic Formula

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.

$$1.5x^{2}-52.5x-4500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.5 for a, -52.5 for b, and -4500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-52.5\right)±\sqrt{\left(-52.5\right)^{2}-4\times 1.5\left(-4500\right)}}{2\times 1.5}$$

Square -52.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-52.5\right)±\sqrt{2756.25-4\times 1.5\left(-4500\right)}}{2\times 1.5}$$

Multiply -4 times 1.5.

$$x=\frac{-\left(-52.5\right)±\sqrt{2756.25-6\left(-4500\right)}}{2\times 1.5}$$

Multiply -6 times -4500.

$$x=\frac{-\left(-52.5\right)±\sqrt{2756.25+27000}}{2\times 1.5}$$

Add 2756.25 to 27000.

$$x=\frac{-\left(-52.5\right)±\sqrt{29756.25}}{2\times 1.5}$$

Take the square root of 29756.25.

$$x=\frac{-\left(-52.5\right)±\frac{345}{2}}{2\times 1.5}$$

The opposite of -52.5 is 52.5.

$$x=\frac{52.5±\frac{345}{2}}{2\times 1.5}$$

Multiply 2 times 1.5.

$$x=\frac{52.5±\frac{345}{2}}{3}$$

Now solve the equation x=\frac{52.5±\frac{345}{2}}{3} when ± is plus. Add 52.5 to \frac{345}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{225}{3}$$

Divide 225 by 3.

$$x=75$$

Now solve the equation x=\frac{52.5±\frac{345}{2}}{3} when ± is minus. Subtract \frac{345}{2} from 52.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{120}{3}$$

Divide -120 by 3.

$$x=-40$$

The equation is now solved.

$$x=75 x=-40$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$1.5x^{2}-52.5x-4500=0$$

Add 4500 to both sides of the equation.

$$1.5x^{2}-52.5x-4500-\left(-4500\right)=-\left(-4500\right)$$

Subtracting -4500 from itself leaves 0.

$$1.5x^{2}-52.5x=-\left(-4500\right)$$

Subtract -4500 from 0.

$$1.5x^{2}-52.5x=4500$$

Divide both sides of the equation by 1.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{1.5x^{2}-52.5x}{1.5}=\frac{4500}{1.5}$$

Dividing by 1.5 undoes the multiplication by 1.5.

$$x^{2}+\left(-\frac{52.5}{1.5}\right)x=\frac{4500}{1.5}$$

Divide -52.5 by 1.5 by multiplying -52.5 by the reciprocal of 1.5.

$$x^{2}-35x=\frac{4500}{1.5}$$

Divide 4500 by 1.5 by multiplying 4500 by the reciprocal of 1.5.

$$x^{2}-35x=3000$$

Divide -35, the coefficient of the x term, by 2 to get -\frac{35}{2}. Then add the square of -\frac{35}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-35x+\left(-\frac{35}{2}\right)^{2}=3000+\left(-\frac{35}{2}\right)^{2}$$

Square -\frac{35}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-35x+\frac{1225}{4}=3000+\frac{1225}{4}$$

Add 3000 to \frac{1225}{4}.

$$x^{2}-35x+\frac{1225}{4}=\frac{13225}{4}$$

Factor x^{2}-35x+\frac{1225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{35}{2}\right)^{2}=\frac{13225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{2}\right)^{2}}=\sqrt{\frac{13225}{4}}$$

Simplify.

$$x-\frac{35}{2}=\frac{115}{2} x-\frac{35}{2}=-\frac{115}{2}$$

Add \frac{35}{2} to both sides of the equation.

$$x=75 x=-40$$