Question

Using the quadratic formula, find the solution to x2+6x+9=0
.(1 point)
Remaining Attempts : 2View previous attempt

Answer 1

Solve −2x2+3x−1=−2
using the quadratic formula. Round the solution(s) to the nearest hundredths. Enter the smaller solution first.(1 point)

Answer 2

To use the quadratic formula, we first need to express both equations in standard form \( ax^2 + bx + c = 0 \).

### First Equation: \(x^2 + 6x + 9 = 0\)

This equation is already in standard form with:
- \( a = 1 \)
- \( b = 6 \)
- \( c = 9 \)

Now, we apply the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Calculating the discriminant (\(b^2 - 4ac\)):
\[
b^2 - 4ac = 6^2 - 4 \cdot 1 \cdot 9 = 36 - 36 = 0
\]

Since the discriminant is 0, there is one real solution:
\[
x = \frac{-6 \pm \sqrt{0}}{2 \cdot 1} = \frac{-6}{2} = -3
\]

### Second Equation: \(-2x^2 + 3x - 1 = -2\)

First, we need to rearrange the equation to standard form:
\[
-2x^2 + 3x - 1 + 2 = 0 \implies -2x^2 + 3x + 1 = 0
\]

Now, multiply the equation by \(-1\) to make \(a\) positive:
\[
2x^2 - 3x - 1 = 0
\]

In this case, we have:
- \( a = 2 \)
- \( b = -3 \)
- \( c = -1 \)

Now, we apply the quadratic formula again:
\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2}
\]

Calculating the discriminant:
\[
(-3)^2 - 4 \cdot 2 \cdot (-1) = 9 + 8 = 17
\]

Now, substituting back into the formula:
\[
x = \frac{3 \pm \sqrt{17}}{4}
\]

Calculating the two solutions:
1. The smaller solution:
\[
x_1 = \frac{3 - \sqrt{17}}{4}
\]

2. The larger solution:
\[
x_2 = \frac{3 + \sqrt{17}}{4}
\]

Now, we round the solutions to the nearest hundredths:
- First, calculate \( \sqrt{17} \approx 4.123 \)
- Then,
\[
x_1 = \frac{3 - 4.123}{4} \approx \frac{-1.123}{4} \approx -0.281
\]
Rounding: \( -0.28 \)

\[
x_2 = \frac{3 + 4.123}{4} \approx \frac{7.123}{4} \approx 1.781
\]
Rounding: \( 1.78 \)

### Final Answers:
1. The solution to \( x^2 + 6x + 9 = 0 \) is \( x = -3 \).
2. The solutions to \(-2x^2 + 3x - 1 = -2\) are approximately \( x_1 \approx -0.28 \) and \( x_2 \approx 1.78 \). Entering the smaller solution first:
- \( -0.28, 1.78 \)