Questions: Solve by completing the square. x^2 + 14x = -19 Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. x= x= or x=

$Solve by completing the square. x^2 + 14x = -19 Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. x= x= or x=$

Transcript text: Solve by completing the square. \[ x^{2}+14 x=-19 \] Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. \[ x= \] $x=$ $\square$ or $x=$ $\square$

Solution

Solution Steps

To solve the quadratic equation by completing the square, we first move the constant term to the other side of the equation. Then, we find the value that completes the square for the quadratic expression on the left side. We add this value to both sides of the equation, rewrite the left side as a squared binomial, and solve for $x$ by taking the square root of both sides.

Step 1: Move the Constant

We start with the equation: \[ x^2 + 14x = -19 \] To complete the square, we first move $-19$ to the left side: \[ x^2 + 14x + 19 = 0 \]

Step 2: Complete the Square

Next, we complete the square for the expression $x^2 + 14x$. We take half of the coefficient of $x$ (which is $14$), square it, and add it to both sides: \[ \left(\frac{14}{2}\right)^2 = 49 \] Thus, we rewrite the equation as: \[ x^2 + 14x + 49 = 49 - 19 \] This simplifies to: \[ (x + 7)^2 = 30 \]

Step 3: Solve for $x$

Now, we take the square root of both sides: \[ x + 7 = \pm \sqrt{30} \] Subtracting $7$ from both sides gives us: \[ x = -7 \pm \sqrt{30} \] This results in two solutions: \[ x = -7 - \sqrt{30} \quad \text{and} \quad x = -7 + \sqrt{30} \]