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Which of the following is a solution to the quadratic equation 2x2−5x=6

? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)
Responses

x=−1.03
x equals negative 1.03

x=−0.89
x equals negative 0.89

x=0.89
x equals 0.89

x=1.03

11 answers

To solve the quadratic equation \( 2x^2 - 5x - 6 = 0 \), we can either factor it, complete the square, or use the quadratic formula. Here, we will use the quadratic formula, which is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For the equation \( 2x^2 - 5x - 6 = 0 \), the coefficients are:

  • \( a = 2 \)
  • \( b = -5 \)
  • \( c = -6 \)

Now, we can calculate the discriminant (\( b^2 - 4ac \)):

\[ b^2 - 4ac = (-5)^2 - 4(2)(-6) = 25 + 48 = 73 \]

Next, substitute the values into the quadratic formula:

\[ x = \frac{-(-5) \pm \sqrt{73}}{2(2)} = \frac{5 \pm \sqrt{73}}{4} \]

Calculating \( \sqrt{73} \):

\[ \sqrt{73} \approx 8.544 \]

Now, we can find the two potential solutions:

  1. \( x = \frac{5 + 8.544}{4} = \frac{13.544}{4} \approx 3.386 \)
  2. \( x = \frac{5 - 8.544}{4} = \frac{-3.544}{4} \approx -0.886 \)

Rounding to the nearest hundredth, we get:

  1. \( x \approx 3.39 \)
  2. \( x \approx -0.89 \)

So, the approximate solutions are \( x \approx 3.39 \) and \( x \approx -0.89 \).

From the given options, the one that matches is:

x = -0.89.

How many real solutions does the following quadratic equation have?

4x2+x+3=0

(1 point)
Responses

two real solutions
two real solutions

no real solutions
no real solutions

three real solutions
three real solutions

one real solution

To determine the number of real solutions to the quadratic equation \(4x^2 + x + 3 = 0\), we can use the discriminant method. The discriminant (\(D\)) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula:

\[ D = b^2 - 4ac \]

For the equation \(4x^2 + x + 3 = 0\), we have:

  • \(a = 4\)
  • \(b = 1\)
  • \(c = 3\)

Now calculate the discriminant:

\[ D = (1)^2 - 4(4)(3) = 1 - 48 = -47 \]

Since the discriminant is negative (\(D < 0\)), this indicates that there are no real solutions to the equation.

Therefore, the correct response is:

no real solutions.

Given the quadratic equation y=5x2+8x+9
, solve for the y-coordinate of the parabola when x=−3
.(1 point)
(−3
,
)

To find the y-coordinate of the parabola represented by the quadratic equation \( y = 5x^2 + 8x + 9 \) when \( x = -3 \), we can substitute \(-3\) into the equation for \(x\):

\[ y = 5(-3)^2 + 8(-3) + 9 \]

Calculating step by step:

  1. Calculate \((-3)^2\): \[ (-3)^2 = 9 \]

  2. Multiply by \(5\): \[ 5 \cdot 9 = 45 \]

  3. Calculate \(8 \cdot (-3)\): \[ 8 \cdot (-3) = -24 \]

  4. Now substitute these values back into the equation: \[ y = 45 - 24 + 9 \]

  5. Perform the addition: \[ 45 - 24 = 21 \] \[ 21 + 9 = 30 \]

Thus, when \( x = -3 \), the y-coordinate is \( 30 \).

The coordinates of the point on the parabola are:

\[ (-3, 30) \]

Given this table of values, complete the table for the quadratic equation y=3x2−6x+9
to find the values of the high or low point of the parabola.

x y
−3
54
−2
33
−1

0
1
2 9
(1 point)
The high or low point of this parabola is (
,
).

To find the high or low point of the parabola given by the quadratic equation \(y = 3x^2 - 6x + 9\), we can calculate its vertex. The vertex of a parabola in the form \(y = ax^2 + bx + c\) can be found using the formula for the x-coordinate of the vertex:

\[ x = -\frac{b}{2a} \]

For the equation \(y = 3x^2 - 6x + 9\):

  • \(a = 3\)
  • \(b = -6\)

Substituting these values into the formula gives us:

\[ x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1 \]

Next, we will substitute \(x = 1\) back into the equation to find the corresponding \(y\)-coordinate:

\[ y = 3(1)^2 - 6(1) + 9 \] \[ y = 3(1) - 6 + 9 = 3 - 6 + 9 = 6 \]

Thus, the vertex, which represents the low point of the parabola (since \(a = 3 > 0\), the parabola opens upwards), is:

\[ (1, 6) \]

Now, let's complete the table for the quadratic equation using the given values:

Completing the Table:

  • For \(x = -3\): \[ y = 3(-3)^2 - 6(-3) + 9 = 3(9) + 18 + 9 = 27 + 18 + 9 = 54 \]
  • For \(x = -2\): \[ y = 3(-2)^2 - 6(-2) + 9 = 3(4) + 12 + 9 = 12 + 12 + 9 = 33 \]
  • For \(x = -1\): \[ y = 3(-1)^2 - 6(-1) + 9 = 3(1) + 6 + 9 = 3 + 6 + 9 = 18 \]
  • For \(x = 0\): \[ y = 3(0)^2 - 6(0) + 9 = 0 + 0 + 9 = 9 \]
  • For \(x = 1\): \[ y = 3(1)^2 - 6(1) + 9 = 3(1) - 6 + 9 = 3 - 6 + 9 = 6 \]
  • For \(x = 2\): \[ y = 3(2)^2 - 6(2) + 9 = 3(4) - 12 + 9 = 12 - 12 + 9 = 9 \]

Final Table:

| x | y | |-----|-----| | -3 | 54 | | -2 | 33 | | -1 | 18 | | 0 | 9 | | 1 | 6 | | 2 | 9 |

Conclusion:

The high or low point of this parabola is \( (1, 6) \).

Use the image to answer the question.

A parabola is graphed on a coordinate plane. The x-axis ranges from negative 25 to 10 in increments of 5. The y-axis ranges from negative 10 to 60 in increments of 10.

Identify the vertex and axis of symmetry of the parabola.

(1 point)
axis of symmetry: x=

vertex: (
)

To determine the vertex and axis of symmetry of a parabola graphed on a coordinate plane, we typically look for:

  1. Vertex: The point at which the parabola changes direction, which is either a high point (maximum) or a low point (minimum).

  2. Axis of Symmetry: A vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. The equation of the axis of symmetry can be represented as \(x = k\), where \(k\) is the x-coordinate of the vertex.

Step-by-Step Approach:

  1. Find the Vertex:

    • Look for the highest or lowest point on the parabola (the peak or trough). The coordinates of that point give you the vertex.

  2. Determine the Axis of Symmetry:

    • The x-coordinate of the vertex is the same for the axis of symmetry. Therefore, if the vertex is \((h, k)\), then the axis of symmetry is \(x = h\).

Example (Hypothetical):

Suppose you found from the graph that:

  • The vertex is at the point \((-5, 20)\).
  • This means the vertex is \((-5, 20)\) and the axis of symmetry would be the line \(x = -5\).

Answer Format:

If you can provide the specific coordinates of the vertex you observe in the image (if you have access to it), I can help you finalize the response with the correct values. However, a typical answer would look like this based on hypothetical coordinates:

  • Axis of symmetry: \(x = -5\)
  • Vertex: \((-5, 20)\)

If you provide the vertex coordinates from the graph, I can help you to finalize the response accordingly!

Which of the following is a point you can generate to create the parabola for the quadratic equation y=−6x2+3x+8
?(1 point)
Responses

(−3,−55)
left parenthesis negative 3 comma negative 55 right parentheis

(−3,323)
left parenthesis negative 3 comma 323 right parentheis

(−6,206)
left parenthesis negative 6 comma 206 right parentheis

(−6,8)

To determine which of the given points lies on the parabola defined by the equation \( y = -6x^2 + 3x + 8 \), we can substitute the x-coordinates of each point into the equation and see if the resulting y-coordinate matches.

Given equation:

\[ y = -6x^2 + 3x + 8 \]

Check each point:

  1. For the point \((-3, -55)\): \[ y = -6(-3)^2 + 3(-3) + 8 \] \[ = -6(9) - 9 + 8 \] \[ = -54 - 9 + 8 \] \[ = -55 \quad \text{(matches)} \]

  2. For the point \((-3, 323)\): (Already checked \(-3\) above, gives \(-55\), so this does not match)

  3. For the point \((-6, 206)\): \[ y = -6(-6)^2 + 3(-6) + 8 \] \[ = -6(36) - 18 + 8 \] \[ = -216 - 18 + 8 \] \[ = -226 \quad \text{(does not match)} \]

  4. For the point \((-6, 8)\): \[ y = -6(-6)^2 + 3(-6) + 8 \] (This calculation is the same as above, which gives \(-226\), so here also does not match.)

Conclusion:

The point that satisfies the equation \( y = -6x^2 + 3x + 8 \) is:

\((-3, -55)\).

So, the correct response is: \((-3, -55)\).