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How many solutions are in the solution set for each quadratic equation below?(3 points)

5x2=50
This quadratic equation would have
solution(s).

3x2=−12
This quadratic equation would have
solution(s).

8x2=0
This quadratic equation would have
solution(s).

5 answers

To determine how many solutions each quadratic equation has, we can analyze each equation.

  1. For the equation \(5x^2 = 50\):

    • First, divide both sides by 5: \[ x^2 = 10 \]
    • The solutions for \(x\) can be found by taking the square root: \[ x = \pm \sqrt{10} \]
    • Since there are two solutions (\(\sqrt{10}\) and \(-\sqrt{10}\)), this quadratic equation would have 2 solution(s).

  2. For the equation \(3x^2 = -12\):

    • First, divide both sides by 3: \[ x^2 = -4 \]
    • Since the right side is negative, there are no real solutions (the square of a real number cannot be negative). Thus, the solutions are complex: \[ x = \pm 2i \]
    • Since there are two complex solutions, this quadratic equation would have 2 solution(s).

  3. For the equation \(8x^2 = 0\):

    • Divide both sides by 8: \[ x^2 = 0 \]
    • The only solution is: \[ x = 0 \]
    • Since there's only one unique solution (counting multiplicity), this quadratic equation would have 1 solution(s).

Summary:

  • \(5x^2 = 50\) would have 2 solution(s).
  • \(3x^2 = -12\) would have 2 solution(s).
  • \(8x^2 = 0\) would have 1 solution(s).
Solve the quadratic equation below using square roots.

(x−4)^2=25


(1 point)
Responses

{−29−−√, 29−−√}
{−29−−√, 29−−√}

{−3, 7}
{−3, 7}

{−2, 6}
{−2, 6}

{−1, 9}

To solve the quadratic equation \((x - 4)^2 = 25\) using square roots, follow these steps:

  1. Take the square root of both sides: \[ x - 4 = \pm 5 \]

  2. This gives us two separate equations to solve:

    • For the positive root: \[ x - 4 = 5 \] \[ x = 5 + 4 = 9 \]

    • For the negative root: \[ x - 4 = -5 \] \[ x = -5 + 4 = -1 \]

  3. Thus, the solutions to the equation are: \[ x = 9 \quad \text{and} \quad x = -1 \]

So the final answer is {-1, 9}.

What should be the first step when solving the quadratic equation below?

(x+10)^2−25=36


(1 point)
Responses

Subtract 10 from both sides of the equation.
Subtract 10 from both sides of the equation.

Add 25 to both sides of the equation.
Add 25 to both sides of the equation.

Take the square root of both sides of the equation.
Take the square root of both sides of the equation.

Divide both sides of the equation by 2.

To solve the quadratic equation \((x + 10)^2 - 25 = 36\), the first step should be to isolate the squared term.

Here’s how you can reason through it:

  1. Start with the equation: \[ (x + 10)^2 - 25 = 36 \]

  2. To isolate \((x + 10)^2\), you should first add 25 to both sides of the equation:

    \[ (x + 10)^2 - 25 + 25 = 36 + 25 \] This simplifies to: \[ (x + 10)^2 = 61 \]

Therefore, the correct response is Add 25 to both sides of the equation.