Solve 0=left(x+1.5right)^2+25

$$0= { \left(x+1.5 \right) }^{ 2 } +25$$

Solve for x (complex solution)

$$x=-1.5-5i$$

$$x=-1.5+5i$$

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}.

$$0=x^{2}+3x+2.25+25$$

Add 2.25 and 25 to get 27.25.

$$0=x^{2}+3x+27.25$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+3x+27.25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and 27.25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 27.25}}{2}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 27.25}}{2}$$

Multiply -4 times 27.25.

$$x=\frac{-3±\sqrt{9-109}}{2}$$

Add 9 to -109.

$$x=\frac{-3±\sqrt{-100}}{2}$$

Take the square root of -100.

$$x=\frac{-3±10i}{2}$$

Now solve the equation x=\frac{-3±10i}{2} when ± is plus. Add -3 to 10i.

$$x=\frac{-3+10i}{2}$$

Divide -3+10i by 2.

$$x=-\frac{3}{2}+5i$$

Now solve the equation x=\frac{-3±10i}{2} when ± is minus. Subtract 10i from -3.

$$x=\frac{-3-10i}{2}$$

Divide -3-10i by 2.

$$x=-\frac{3}{2}-5i$$

The equation is now solved.

$$x=-\frac{3}{2}+5i x=-\frac{3}{2}-5i$$

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}.

$$0=x^{2}+3x+2.25+25$$

Add 2.25 and 25 to get 27.25.

$$0=x^{2}+3x+27.25$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+3x+27.25=0$$

Subtract 27.25 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+3x=-27.25$$

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.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=-27.25+\left(\frac{3}{2}\right)^{2}$$

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

$$x^{2}+3x+\frac{9}{4}=\frac{-109+9}{4}$$

Add -27.25 to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3x+\frac{9}{4}=-25$$

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}.

$$\left(x+\frac{3}{2}\right)^{2}=-25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{-25}$$

Simplify.

$$x+\frac{3}{2}=5i x+\frac{3}{2}=-5i$$

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

$$x=-\frac{3}{2}+5i x=-\frac{3}{2}-5i$$