Solve 15x^2+19.5x+4.5=0

$$15 x ^ { 2 } + 19.5 x + 4.5 = 0$$

Solve for x

$$x=-0.3$$

$$x=-1$$

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.

$$15x^{2}+19.5x+4.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, 19.5 for b, and 4.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19.5±\sqrt{19.5^{2}-4\times 15\times 4.5}}{2\times 15}$$

Square 19.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-19.5±\sqrt{380.25-4\times 15\times 4.5}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-19.5±\sqrt{380.25-60\times 4.5}}{2\times 15}$$

Multiply -60 times 4.5.

$$x=\frac{-19.5±\sqrt{380.25-270}}{2\times 15}$$

Add 380.25 to -270.

$$x=\frac{-19.5±\sqrt{110.25}}{2\times 15}$$

Take the square root of 110.25.

$$x=\frac{-19.5±\frac{21}{2}}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{-19.5±\frac{21}{2}}{30}$$

Now solve the equation x=\frac{-19.5±\frac{21}{2}}{30} when ± is plus. Add -19.5 to \frac{21}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{9}{30}$$

Reduce the fraction \frac{-9}{30} to lowest terms by extracting and canceling out 3.

$$x=-\frac{3}{10}$$

Now solve the equation x=\frac{-19.5±\frac{21}{2}}{30} when ± is minus. Subtract \frac{21}{2} from -19.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{30}{30}$$

Divide -30 by 30.

$$x=-1$$

The equation is now solved.

$$x=-\frac{3}{10} x=-1$$

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.

$$15x^{2}+19.5x+4.5=0$$

Subtract 4.5 from both sides of the equation.

$$15x^{2}+19.5x+4.5-4.5=-4.5$$

Subtracting 4.5 from itself leaves 0.

$$15x^{2}+19.5x=-4.5$$

Divide both sides by 15.

$$\frac{15x^{2}+19.5x}{15}=-\frac{4.5}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$x^{2}+\frac{19.5}{15}x=-\frac{4.5}{15}$$

Divide 19.5 by 15.

$$x^{2}+1.3x=-\frac{4.5}{15}$$

Divide -4.5 by 15.

$$x^{2}+1.3x=-0.3$$

Divide 1.3, the coefficient of the x term, by 2 to get 0.65. Then add the square of 0.65 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+1.3x+0.65^{2}=-0.3+0.65^{2}$$

Square 0.65 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+1.3x+0.4225=-0.3+0.4225$$

Add -0.3 to 0.4225 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+1.3x+0.4225=0.1225$$

Factor x^{2}+1.3x+0.4225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+0.65\right)^{2}=0.1225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+0.65\right)^{2}}=\sqrt{0.1225}$$

Simplify.

$$x+0.65=\frac{7}{20} x+0.65=-\frac{7}{20}$$

Subtract 0.65 from both sides of the equation.

$$x=-\frac{3}{10} x=-1$$