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Complete the square to solve the following quadratic equation: x^2+2x−8=0.(1 point)

Responses

x=10, x=−8
x equals 10 , x equals negative 8

x=2, x=−4
x equals 2 , x equals negative 4

x=−10, x=8
x equals negative 10 , x equals 8

x=−2, x=4

9 answers

To complete the square for the quadratic equation \( x^2 + 2x - 8 = 0 \), we follow these steps:

  1. Move the constant term to the other side of the equation: \[ x^2 + 2x = 8 \]

  2. Take half of the coefficient of \( x \), square it, and add it to both sides. The coefficient of \( x \) is 2, so half of that is 1, and squaring it gives us 1: \[ x^2 + 2x + 1 = 8 + 1 \]

  3. Now, the left side is a perfect square: \[ (x + 1)^2 = 9 \]

  4. Take the square root of both sides: \[ x + 1 = 3 \quad \text{or} \quad x + 1 = -3 \]

  5. Solve for \( x \): \[ x = 3 - 1 = 2 \quad \text{or} \quad x = -3 - 1 = -4 \]

Thus, the solutions are \( x = 2 \) and \( x = -4 \).

The correct response is: x=2, x=−4.

Complete the square to solve the following quadratic equation: x^2−22x+1=15 .(1 point)
Responses

x=11+315−−√, 11−315−−√
x equals 11 plus 3 Start Root 15 End Root comma 11 minus 3 Start Root 15 End Root

x=−11+14−−√, −11−14−−√
x equals negative 11 plus Start Root 14 End Root comma negative 11 minus Start Root 14 End Root

x=11+14−−√, 11−14−−√
x equals 11 plus Start Root 14 End Root comma 11 minus Start Root 14 End Root

x=−11+315−−√, −11−315−−√

To solve the equation \( x^2 - 22x + 1 = 15 \) by completing the square, we need to follow these steps:

  1. Move the constant from the right side: \[ x^2 - 22x + 1 - 15 = 0 \] Simplifying gives: \[ x^2 - 22x - 14 = 0 \]

  2. Rearrange the equation: \[ x^2 - 22x = 14 \]

  3. Complete the square:

    • Take half of \(-22\) (which is \(-11\)) and square it (which gives \(121\)).
    • Add \(121\) to both sides: \[ x^2 - 22x + 121 = 14 + 121 \] Simplifying gives: \[ (x - 11)^2 = 135 \]

  4. Take the square root of both sides: \[ x - 11 = \sqrt{135} \quad \text{or} \quad x - 11 = -\sqrt{135} \]

  5. Solve for \( x \): \[ x = 11 + \sqrt{135} \quad \text{or} \quad x = 11 - \sqrt{135} \]

Since \( \sqrt{135} = 3\sqrt{15} \), we can rewrite the solutions as: \[ x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15} \]

So the correct response is: x=11+3√15, 11−3√15.

Using the quadratic formula, find the solution to x^2−6x−16=0.(1 point)
Responses

x=−2
x equals negative 2

x=8, x=−2
x equals 8 , x equals negative 2

x=8
x equals 8

x=8, x=−8

To solve the quadratic equation \( x^2 - 6x - 16 = 0 \) using the quadratic formula, we use the formula:

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

where \( a = 1 \), \( b = -6 \), and \( c = -16 \).

  1. Identify coefficients: \[ a = 1, \quad b = -6, \quad c = -16 \]

  2. Calculate the discriminant: \[ b^2 - 4ac = (-6)^2 - 4 \cdot 1 \cdot (-16) = 36 + 64 = 100 \]

  3. Apply the quadratic formula: \[ x = \frac{-(-6) \pm \sqrt{100}}{2 \cdot 1} = \frac{6 \pm 10}{2} \]

  4. Calculate the two possible solutions:

    • For the positive square root: \[ x = \frac{6 + 10}{2} = \frac{16}{2} = 8 \]
    • For the negative square root: \[ x = \frac{6 - 10}{2} = \frac{-4}{2} = -2 \]

Thus, the solutions are \( x = 8 \) and \( x = -2 \).

The correct response is: x=8, x=−2.

Which of the following is a solution to the quadratic equation 2x^2−5x=6 ? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)
Responses

x=0.89
x equals 0.89

x=1.03
x equals 1.03

x=−0.89
x equals negative 0.89

x=−1.03

To solve the quadratic equation \( 2x^2 - 5x = 6 \), we first rearrange it into standard form:

\[ 2x^2 - 5x - 6 = 0 \]

Next, we can apply the quadratic formula:

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

where \( a = 2 \), \( b = -5 \), and \( c = -6 \).

  1. Identify coefficients: \[ a = 2, \quad b = -5, \quad c = -6 \]

  2. Calculate the discriminant: \[ b^2 - 4ac = (-5)^2 - 4 \cdot 2 \cdot (-6) = 25 + 48 = 73 \]

  3. Apply the quadratic formula: \[ x = \frac{-(-5) \pm \sqrt{73}}{2 \cdot 2} = \frac{5 \pm \sqrt{73}}{4} \]

  4. Calculate the two possible solutions:

    • For the positive square root: \[ x = \frac{5 + \sqrt{73}}{4} \]
    • For the negative square root: \[ x = \frac{5 - \sqrt{73}}{4} \]

  5. Approximate the values: First, we calculate \( \sqrt{73} \approx 8.544 \).

    • For the positive solution: \[ x \approx \frac{5 + 8.544}{4} \approx \frac{13.544}{4} \approx 3.386 \]

    • For the negative solution: \[ x \approx \frac{5 - 8.544}{4} \approx \frac{-3.544}{4} \approx -0.886 \]

Rounding to the nearest hundredth, the approximate solutions are:

  • \( x \approx 3.39 \) (positive solution)
  • \( x \approx -0.89 \) (negative solution)

Since the provided options include \( x = -0.89 \) and this is a rounded form of the calculated negative solution, the correct response is:

x=−0.89.

How many real solutions does the following quadratic equation have?

4x^2+x+3=0

(1 point)
Responses

three real solutions
three real solutions

one real solution
one real solution

no real solutions
no real solutions

two real solutions

To determine the number of real solutions for the quadratic equation \(4x^2 + x + 3 = 0\), we can calculate the discriminant (\(D\)). The discriminant is given by the formula:

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

where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).

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

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

Now, we can calculate the discriminant:

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

Since the discriminant \(D\) is negative (\(-47 < 0\)), it indicates that the quadratic equation has no real solutions.

Thus, the correct response is:

no real solutions.