Solve the following quadratic equation by completing the square method.x^2+3.9x+3.5=0. Explain in a simple way pls.

Answer 1

To solve the quadratic equation \( x^2 + 3.9x + 3.5 = 0 \) by completing the square, follow these steps:

Step 1: Move the constant to the other side.

Start with the original equation: \[ x^2 + 3.9x + 3.5 = 0 \] Subtract \( 3.5 \) from both sides: \[ x^2 + 3.9x = -3.5 \]

Step 2: Complete the square.

To complete the square, you need to transform the left side of the equation into a perfect square trinomial. Here's how you do it:

1. Take the coefficient of \( x \) (which is \( 3.9 \)), divide it by 2, and square it: \[ \left(\frac{3.9}{2}\right)^2 = \left(1.95\right)^2 = 3.8025 \]

2. Add this number to both sides of the equation: \[ x^2 + 3.9x + 3.8025 = -3.5 + 3.8025 \] This simplifies to: \[ x^2 + 3.9x + 3.8025 = 0.3025 \]

Step 3: Rewrite the left side as a squared term.

Now, the left side can be factored as: \[ (x + 1.95)^2 = 0.3025 \]

Step 4: Take the square root of both sides.

Now we take the square root of both sides. Don't forget to consider both the positive and negative square roots: \[ x + 1.95 = \pm \sqrt{0.3025} \] Calculating the square root: \[ \sqrt{0.3025} = 0.55 \] So we have: \[ x + 1.95 = 0.55 \quad \text{or} \quad x + 1.95 = -0.55 \]

Step 5: Solve for x.

Now we solve for \( x \) in both cases:

1. For \( x + 1.95 = 0.55 \): \[ x = 0.55 - 1.95 = -1.4 \]

2. For \( x + 1.95 = -0.55 \): \[ x = -0.55 - 1.95 = -2.5 \]

Final Solutions

The solutions to the quadratic equation \( x^2 + 3.9x + 3.5 = 0 \) are: \[ x = -1.4 \quad \text{and} \quad x = -2.5 \]