Solve (x+1.5)^2=2.25

$$( x + 1.5 ) ^ { 2 } = 2.25$$

Solve for x

$$x=0$$

$$x=-3$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1.5\right)^{2}.

$$x^{2}+3x+2.25=2.25$$

Subtract 2.25 from both sides.

$$x^{2}+3x+2.25-2.25=0$$

Subtract 2.25 from 2.25 to get 0.

$$x^{2}+3x=0$$

Factor out x.

$$x\left(x+3\right)=0$$

To find equation solutions, solve x=0 and x+3=0.

$$x=0 x=-3$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1.5\right)^{2}.

$$x^{2}+3x+2.25=2.25$$

Subtract 2.25 from both sides.

$$x^{2}+3x+2.25-2.25=0$$

Subtract 2.25 from 2.25 to get 0.

$$x^{2}+3x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}}}{2}$$

Take the square root of 3^{2}.

$$x=\frac{-3±3}{2}$$

Now solve the equation x=\frac{-3±3}{2} when ± is plus. Add -3 to 3.

$$x=\frac{0}{2}$$

Divide 0 by 2.

$$x=0$$

Now solve the equation x=\frac{-3±3}{2} when ± is minus. Subtract 3 from -3.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

The equation is now solved.

$$x=0 x=-3$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1.5\right)^{2}.

$$x^{2}+3x+2.25=2.25$$

Subtract 2.25 from both sides.

$$x^{2}+3x+2.25-2.25=0$$

Subtract 2.25 from 2.25 to get 0.

$$x^{2}+3x=0$$

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=\frac{9}{4}$$

Factor x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{3}{2} x+\frac{3}{2}=-\frac{3}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=0 x=-3$$