Solve x^2+150x+5609=0

$${ x }^{ 2 } +150x+5609=0$$

Solve for x

$$x=-79$$

$$x=-71$$

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.

$$x^{2}+150x+5609=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 150 for b, and 5609 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-150±\sqrt{150^{2}-4\times 5609}}{2}$$

Square 150.

$$x=\frac{-150±\sqrt{22500-4\times 5609}}{2}$$

Multiply -4 times 5609.

$$x=\frac{-150±\sqrt{22500-22436}}{2}$$

Add 22500 to -22436.

$$x=\frac{-150±\sqrt{64}}{2}$$

Take the square root of 64.

$$x=\frac{-150±8}{2}$$

Now solve the equation x=\frac{-150±8}{2} when ± is plus. Add -150 to 8.

$$x=-\frac{142}{2}$$

Divide -142 by 2.

$$x=-71$$

Now solve the equation x=\frac{-150±8}{2} when ± is minus. Subtract 8 from -150.

$$x=-\frac{158}{2}$$

Divide -158 by 2.

$$x=-79$$

The equation is now solved.

$$x=-71 x=-79$$

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.

$$x^{2}+150x+5609=0$$

Subtract 5609 from both sides of the equation.

$$x^{2}+150x+5609-5609=-5609$$

Subtracting 5609 from itself leaves 0.

$$x^{2}+150x=-5609$$

Divide 150, the coefficient of the x term, by 2 to get 75. Then add the square of 75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+150x+75^{2}=-5609+75^{2}$$

Square 75.

$$x^{2}+150x+5625=-5609+5625$$

Add -5609 to 5625.

$$x^{2}+150x+5625=16$$

Factor x^{2}+150x+5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+75\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+75\right)^{2}}=\sqrt{16}$$

Simplify.

$$x+75=4 x+75=-4$$

Subtract 75 from both sides of the equation.

$$x=-71 x=-79$$