Solve 25x^2+150x+225=0

$$25 { x }^{ 2 } +150x+225=0$$

Solve for x

$$x=-3$$

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.

$$25x^{2}+150x+225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, 150 for b, and 225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-150±\sqrt{150^{2}-4\times 25\times 225}}{2\times 25}$$

Square 150.

$$x=\frac{-150±\sqrt{22500-4\times 25\times 225}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-150±\sqrt{22500-100\times 225}}{2\times 25}$$

Multiply -100 times 225.

$$x=\frac{-150±\sqrt{22500-22500}}{2\times 25}$$

Add 22500 to -22500.

$$x=\frac{-150±\sqrt{0}}{2\times 25}$$

Take the square root of 0.

$$x=-\frac{150}{2\times 25}$$

Multiply 2 times 25.

$$x=-\frac{150}{50}$$

Divide -150 by 50.

$$x=-3$$

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.

$$25x^{2}+150x+225=0$$

Subtract 225 from both sides of the equation.

$$25x^{2}+150x+225-225=-225$$

Subtracting 225 from itself leaves 0.

$$25x^{2}+150x=-225$$

Divide both sides by 25.

$$\frac{25x^{2}+150x}{25}=-\frac{225}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$x^{2}+\frac{150}{25}x=-\frac{225}{25}$$

Divide 150 by 25.

$$x^{2}+6x=-\frac{225}{25}$$

Divide -225 by 25.

$$x^{2}+6x=-9$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=-9+3^{2}$$

Square 3.

$$x^{2}+6x+9=-9+9$$

Add -9 to 9.

$$x^{2}+6x+9=0$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{0}$$

Simplify.

$$x+3=0 x+3=0$$

Subtract 3 from both sides of the equation.

$$x=-3 x=-3$$

The equation is now solved. Solutions are the same.

$$x=-3$$