Solve 1.5x+0.1x^2=45

$$1.5 x + 0.1 x ^ { 2 } = 45$$

Solve for x

$$x=-30$$

$$x=15$$

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.

$$0.1x^{2}+1.5x=45$$

Subtract 45 from both sides of the equation.

$$0.1x^{2}+1.5x-45=45-45$$

Subtracting 45 from itself leaves 0.

$$0.1x^{2}+1.5x-45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.1 for a, 1.5 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1.5±\sqrt{1.5^{2}-4\times 0.1\left(-45\right)}}{2\times 0.1}$$

Square 1.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-1.5±\sqrt{2.25-4\times 0.1\left(-45\right)}}{2\times 0.1}$$

Multiply -4 times 0.1.

$$x=\frac{-1.5±\sqrt{2.25-0.4\left(-45\right)}}{2\times 0.1}$$

Multiply -0.4 times -45.

$$x=\frac{-1.5±\sqrt{2.25+18}}{2\times 0.1}$$

Add 2.25 to 18.

$$x=\frac{-1.5±\sqrt{20.25}}{2\times 0.1}$$

Take the square root of 20.25.

$$x=\frac{-1.5±\frac{9}{2}}{2\times 0.1}$$

Multiply 2 times 0.1.

$$x=\frac{-1.5±\frac{9}{2}}{0.2}$$

Now solve the equation x=\frac{-1.5±\frac{9}{2}}{0.2} when ± is plus. Add -1.5 to \frac{9}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{3}{0.2}$$

Divide 3 by 0.2 by multiplying 3 by the reciprocal of 0.2.

$$x=15$$

Now solve the equation x=\frac{-1.5±\frac{9}{2}}{0.2} when ± is minus. Subtract \frac{9}{2} from -1.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{6}{0.2}$$

Divide -6 by 0.2 by multiplying -6 by the reciprocal of 0.2.

$$x=-30$$

The equation is now solved.

$$x=15 x=-30$$

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.

$$0.1x^{2}+1.5x=45$$

Multiply both sides by 10.

$$\frac{0.1x^{2}+1.5x}{0.1}=\frac{45}{0.1}$$

Dividing by 0.1 undoes the multiplication by 0.1.

$$x^{2}+\frac{1.5}{0.1}x=\frac{45}{0.1}$$

Divide 1.5 by 0.1 by multiplying 1.5 by the reciprocal of 0.1.

$$x^{2}+15x=\frac{45}{0.1}$$

Divide 45 by 0.1 by multiplying 45 by the reciprocal of 0.1.

$$x^{2}+15x=450$$

Divide 15, the coefficient of the x term, by 2 to get \frac{15}{2}. Then add the square of \frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=450+\left(\frac{15}{2}\right)^{2}$$

Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+15x+\frac{225}{4}=450+\frac{225}{4}$$

Add 450 to \frac{225}{4}.

$$x^{2}+15x+\frac{225}{4}=\frac{2025}{4}$$

Factor x^{2}+15x+\frac{225}{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}.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{2025}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{2025}{4}}$$

Simplify.

$$x+\frac{15}{2}=\frac{45}{2} x+\frac{15}{2}=-\frac{45}{2}$$

Subtract \frac{15}{2} from both sides of the equation.

$$x=15 x=-30$$