Verified Answer :

  • Identify the quadratic equation \(4x^2 - 20x + 25 = 0\).
  • Recognize the expression as a perfect square trinomial \((2x - 5)^2 = 0\).
  • Solve the equation \((2x - 5)^2 = 0\) to find the root \(x = \frac{5}{2}\).
  • The final answer is \(\boxed{\frac{5}{2}}\).

Explanation

  1. Problem Analysis

The given equation is \(4x^2 - 20x + 25 = 0\). This is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 4\), \(b = -20\), and \(c = 25\). We can attempt to solve this by factoring.

  1. Factoring the Expression

To factor the quadratic expression \(4x^2 - 20x + 25\), we look for a pattern, such as a perfect square trinomial \((mx - n)^2 = m^2x^2 - 2mnx + n^2\).
Notice that:

  • The first term is \(4x^2 = (2x)^2\), so \(m = 2\).
  • The last term is \(25 = 5^2\), so \(n = 5\).
  • The middle term should be \(-2mnx = -2(2x)(5) = -20x\).

Since the middle term of our expression matches \(-20x\), the expression is indeed a perfect square trinomial:
\[4x^2 - 20x + 25 = (2x - 5)^2.\]

  1. Solving for x

Now substitute the factored form back into the equation:
\[(2x - 5)^2 = 0.\]
Taking the square root of both sides, we get:
\[2x - 5 = 0.\]
To solve for \(x\), add 5 to both sides:
\[2x = 5.\]
Finally, divide by 2:
\[x = \frac{5}{2}.\]
Thus, the equation has a single real root \(x = 2.5\).

Examples

Solving quadratic equations is fundamental in fields like physics and engineering. For example, if you are modeling the trajectory of a ball, the height \(h\) at time \(t\) can be given by a quadratic equation \(h(t) = -gt^2 + v_0t + h_0\). By setting \(h(t) = 0\), you can find the time at which the ball hits the ground by solving a quadratic equation similar to the one we just solved. Understanding these techniques allows you to accurately predict physical outcomes in various scenarios.

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