Answered

Determine the number of real solutions each quadratic equation has.



$y=12 x^2-9 x+4$

□ real solution(s)



$10 x+y=-x^2+2$

□ real solution(s)



$4 y-7=5 x^2-x+2+3 y$

□ real solution(s)



$y=(-x+4)^2$

□ real solution(s)

Verified Answer :

  • Analyze each quadratic equation by setting it to the standard form \(ax^2 + bx + c = 0\) with \(y=0\).
  • Calculate the discriminant \(D = b^2 - 4ac\) for each equation.
  • Determine the number of real solutions: 0 if \(D < 0\), 1 if \(D = 0\), and 2 if \(D > 0\).
  • The final counts are: \(\boxed{0, 2, 0, 1}\) real solutions respectively for the four equations.

Explanation

  1. Analysis and Method for Determining Real Solutions

To determine the number of real solutions for each quadratic equation, we set \(y=0\) and write each equation in the standard form \(ax^2 + bx + c = 0\). The discriminant \(D = b^2 - 4ac\) determines the nature and number of real solutions:

  • If \(D > 0\), there are two distinct real solutions.
  • If \(D = 0\), there is exactly one real solution.
  • If \(D < 0\), there are no real solutions.

For the given equations:

  1. \(12x^2 - 9x + 4 = 0\)
  2. \(x^2 + 10x - 2 = 0\)
  3. \(5x^2 - x + 9 = 0\)
  4. \((-x + 4)^2 = 0\), which expands to \(x^2 - 8x + 16 = 0\).
  1. Calculating Discriminants and Determining the Number of Solutions

We now calculate the discriminant \(D = b^2 - 4ac\) for each equation:

  1. For \(12x^2 - 9x + 4 = 0\):

\(D = (-9)^2 - 4(12)(4) = 81 - 192 = -111\).
Since \(D < 0\), there are \(0\) real solutions.

  1. For \(x^2 + 10x - 2 = 0\):

\(D = (10)^2 - 4(1)(-2) = 100 + 8 = 108\).
Since \(D > 0\), there are \(2\) real solutions.

  1. For \(5x^2 - x + 9 = 0\):

\(D = (-1)^2 - 4(5)(9) = 1 - 180 = -179\).
Since \(D < 0\), there are \(0\) real solutions.

  1. For \(x^2 - 8x + 16 = 0\):

\(D = (-8)^2 - 4(1)(16) = 64 - 64 = 0\).
Since \(D = 0\), there is \(1\) real solution.

  1. Conclusion and Final Answer

Summarizing our results based on the discriminants calculated:

  1. The equation \(12x^2 - 9x + 4 = 0\) has \(\boxed{0}\) real solutions.
  2. The equation \(x^2 + 10x - 2 = 0\) has \(\boxed{2}\) real solutions.
  3. The equation \(5x^2 - x + 9 = 0\) has \(\boxed{0}\) real solutions.
  4. The equation \(x^2 - 8x + 16 = 0\) has \(\boxed{1}\) real solution.

Examples

In physics and engineering, the discriminant of a quadratic equation is vital for determining the nature of a system's behavior. For example, when analyzing the motion of an object under a constant force, the quadratic equation describing its position relative to time can tell us if and when the object reaches a certain point (like the ground). A positive discriminant indicates two moments in time, a zero discriminant indicates a single moment (like touching the ground at its peak), and a negative discriminant suggests the object never reaches that point. This tool allows engineers to quickly determine the feasibility and timing of physical events.

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