Solved on Feb 08, 2024

Find the leftmost $x$ value by completing the square for $2x^2 + 8x + 6$ . Round your answer to the nearest 0.001.

STEP 1

Assumptions
1. The given quadratic expression is $2x^{2} + 8x + 6$ .
2. Completing the square involves rewriting the quadratic in the form $(x + p)^{2} + q$ .
3. We are looking for the left-most value of $x$ after completing the square, which corresponds to the vertex of the parabola represented by the quadratic equation.
4. Rounding the final answer to the nearest third decimal place.

STEP 2

To complete the square, we need to express the quadratic expression in the form $a(x + b)^{2} + c$ . First, factor out the coefficient of $x^{2}$ from the first two terms.
$2(x^{2} + 4x) + 6$

STEP 3

Next, we find the value that needs to be added and subtracted to complete the square. This value is $\left(\frac{b}{2a}\right)^{2}$ , where $a$ is the coefficient of $x^{2}$ and $b$ is the coefficient of $x$ .
$\left(\frac{4}{2 \cdot 1}\right)^{2} = \left(\frac{4}{2}\right)^{2} = 2^{2} = 4$

STEP 4

Add and subtract the value found in STEP_3 inside the parentheses. We multiply the value by $a$ before subtracting it outside the parentheses to keep the expression equivalent to the original.
$2\left(x^{2} + 4x + 4 - 4\right) + 6$

STEP 5

Rewrite the expression by combining the constant terms inside and outside the parentheses.
$2\left((x + 2)^{2} - 4\right) + 6$

STEP 6

Distribute the $2$ to the terms inside the parentheses.
$2(x + 2)^{2} - 8 + 6$

STEP 7

Combine the constant terms to simplify the expression.
$2(x + 2)^{2} - 2$

STEP 8

Now that we have completed the square, we can identify the vertex of the parabola. The vertex form of a parabola is $a(x - h)^{2} + k$ where $(h, k)$ is the vertex. In our case, the vertex is $(-2, -2)$ .

STEP 9

The left-most value of $x$ corresponds to the $x$ -coordinate of the vertex, which is $-2$ .

STEP 10

Since no rounding is needed for the exact value of $-2$ , the final answer is:
The left-most value of $x$ is $-2$ .

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Find the leftmost $x$ value by completing the square for $2x^2 + 8x + 6$ . Round your answer to the nearest 0.001.

STEP 1

STEP 2

To complete the square, we need to express the quadratic expression in the form $a(x + b)^{2} + c$ . First, factor out the coefficient of $x^{2}$ from the first two terms.
$2(x^{2} + 4x) + 6$

STEP 3

Next, we find the value that needs to be added and subtracted to complete the square. This value is $\left(\frac{b}{2a}\right)^{2}$ , where $a$ is the coefficient of $x^{2}$ and $b$ is the coefficient of $x$ .
$\left(\frac{4}{2 \cdot 1}\right)^{2} = \left(\frac{4}{2}\right)^{2} = 2^{2} = 4$

STEP 4

Add and subtract the value found in STEP_3 inside the parentheses. We multiply the value by $a$ before subtracting it outside the parentheses to keep the expression equivalent to the original.
$2\left(x^{2} + 4x + 4 - 4\right) + 6$

STEP 5

Rewrite the expression by combining the constant terms inside and outside the parentheses.
$2\left((x + 2)^{2} - 4\right) + 6$

STEP 6

Distribute the $2$ to the terms inside the parentheses.
$2(x + 2)^{2} - 8 + 6$

STEP 7

Combine the constant terms to simplify the expression.
$2(x + 2)^{2} - 2$

STEP 8

Now that we have completed the square, we can identify the vertex of the parabola. The vertex form of a parabola is $a(x - h)^{2} + k$ where $(h, k)$ is the vertex. In our case, the vertex is $(-2, -2)$ .

STEP 9

The left-most value of $x$ corresponds to the $x$ -coordinate of the vertex, which is $-2$ .

STEP 10

Since no rounding is needed for the exact value of $-2$ , the final answer is:
The left-most value of $x$ is $-2$ .

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Find the leftmost xxx value by completing the square for 2x2+8x+62x^2 + 8x + 62x2+8x+6. Round your answer to the nearest 0.001.

STEP 1

STEP 2

To complete the square, we need to express the quadratic expression in the form a(x+b)2+ca(x + b)^{2} + ca(x+b)2+c. First, factor out the coefficient of x2x^{2}x2 from the first two terms.2(x2+4x)+62(x^{2} + 4x) + 62(x2+4x)+6

STEP 3

STEP 4

Add and subtract the value found in STEP_3 inside the parentheses. We multiply the value by aaa before subtracting it outside the parentheses to keep the expression equivalent to the original.2(x2+4x+4−4)+62\left(x^{2} + 4x + 4 - 4\right) + 62(x2+4x+4−4)+6

STEP 5

Rewrite the expression by combining the constant terms inside and outside the parentheses.2((x+2)2−4)+62\left((x + 2)^{2} - 4\right) + 62((x+2)2−4)+6

STEP 6

Distribute the 222 to the terms inside the parentheses.2(x+2)2−8+62(x + 2)^{2} - 8 + 62(x+2)2−8+6

STEP 7

Combine the constant terms to simplify the expression.2(x+2)2−22(x + 2)^{2} - 22(x+2)2−2

STEP 8

Now that we have completed the square, we can identify the vertex of the parabola. The vertex form of a parabola is a(x−h)2+ka(x - h)^{2} + ka(x−h)2+k where (h,k)(h, k)(h,k) is the vertex. In our case, the vertex is (−2,−2)(-2, -2)(−2,−2).

STEP 9

The left-most value of xxx corresponds to the xxx-coordinate of the vertex, which is −2-2−2.

STEP 10

Since no rounding is needed for the exact value of −2-2−2, the final answer is:The left-most value of xxx is −2-2−2.

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Find the leftmost $x$ value by completing the square for $2x^2 + 8x + 6$ . Round your answer to the nearest 0.001.

To complete the square, we need to express the quadratic expression in the form $a(x + b)^{2} + c$ . First, factor out the coefficient of $x^{2}$ from the first two terms.
$2(x^{2} + 4x) + 6$

Add and subtract the value found in STEP_3 inside the parentheses. We multiply the value by $a$ before subtracting it outside the parentheses to keep the expression equivalent to the original.
$2\left(x^{2} + 4x + 4 - 4\right) + 6$

Rewrite the expression by combining the constant terms inside and outside the parentheses.
$2\left((x + 2)^{2} - 4\right) + 6$

Distribute the $2$ to the terms inside the parentheses.
$2(x + 2)^{2} - 8 + 6$

Combine the constant terms to simplify the expression.
$2(x + 2)^{2} - 2$

Now that we have completed the square, we can identify the vertex of the parabola. The vertex form of a parabola is $a(x - h)^{2} + k$ where $(h, k)$ is the vertex. In our case, the vertex is $(-2, -2)$ .

The left-most value of $x$ corresponds to the $x$ -coordinate of the vertex, which is $-2$ .

Since no rounding is needed for the exact value of $-2$ , the final answer is:
The left-most value of $x$ is $-2$ .