IIT Madras
Mathematics 1
Solution of quadratic equation using Square method

Solution of quadratic equation using Square method

A well-defined collection of distinct objects called elements or members.

https://youtu.be/qvUX-jgtogg

Exercise Questions 🤯

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Hello! On this Wednesday evening here in India, I’d be happy to explain these questions. They all revolve around a very powerful technique for working with quadratic equations called “Completing the Square”.

Core Concept: Completing the Square

The goal of “completing the square” is to take a quadratic expression like $x^2 + bx$ and turn it into a perfect square trinomial, which can be factored into $(x+k)^2$.

How to do it:

  1. Take the coefficient of the $x$ term, which is ‘$b$’.
  2. Divide it by 2: $\frac{b}{2}$.
  3. Square the result: $\left(\frac{b}{2}\right)^2$.
  4. This is the “magic number” you need to add to complete the square.

For example, to complete the square for $x^2 + 10x$:

  • $b = 10$.
  • Half of b is $\frac{10}{2} = 5$.
  • Square the result: $5^2 = 25$.
  • So, $x^2 + 10x + 25$ is a perfect square, which factors to $(x+5)^2$.

Let’s apply this method to your questions.

Question 1: Making a Perfect Square (from file image_ff95a5.png)

The Question: What should be added in $p(x)$ to make it perfect square, if $p(x) = x^2 - 12x + 34$?

  • 1
  • 2
  • 3
  • 4

Core Concept: We want to adjust the constant term in the expression so that the entire expression becomes a perfect square trinomial.

Detailed Solution:

  1. Focus on the first two terms: $x^2 - 12x$.
  2. Identify the ‘b’ coefficient: Here, $b = -12$.
  3. Find the number needed to complete the square:
    • Take half of b: $\frac{-12}{2} = -6$.
    • Square the result: $(-6)^2 = 36$.
  4. Analyze the expression: The perfect square trinomial we want is $x^2 - 12x + 36$, which factors to $(x-6)^2$.
  5. Our current expression is $x^2 - 12x + 34$.
  6. To turn the constant term from 34 into the 36 we need, we must add the difference: $36 - 34 = 2$.

Final Answer: You should add 2 to the expression.

Question 2: Solving a Quadratic Equation (from file image_ff95a5.png)

The Question: Choose the correct option regarding equation $x^2 - 12x + 37 = 0$.

  • Can be solved using the prime factorization method
  • 1 is a root
  • Can not be solved by completing the square method
  • 0 is a root

Core Concept: We can use the method of completing the square to solve quadratic equations. If this process leads to taking the square root of a negative number, the equation has no real solutions.

Detailed Solution:

  1. Start with the equation: $x^2 - 12x + 37 = 0$
  2. Move the constant term to the right side: $x^2 - 12x = -37$
  3. Complete the square on the left side: As we found in the previous question, we need to add (-12/2)² = 36 to both sides to make the left side a perfect square. $x^2 - 12x + 36 = -37 + 36$
  4. Factor the left side and simplify the right side: $(x - 6)^2 = -1$
  5. Analyze the result: This equation asks for a number, $(x-6)$, which when squared equals -1. The square of any real number is always non-negative (0 or positive). Therefore, there is no real number solution for this equation.

Evaluating the Options:

  • “Can be solved using the prime factorization method”: This method typically works for integer roots. Since there are no real roots, this is false.
  • “1 is a root” / “0 is a root”: Both are false.
  • “Can not be solved by completing the square method”: This is a poorly worded but common statement. The method of completing the square works perfectly to show us that there are no real solutions. The intended meaning is that the equation cannot be solved to find a real root. Given the other options are clearly false, this is the intended answer.

Final Answer: Can not be solved by completing the square method (meaning it cannot be solved for real roots).

Question 3: Solving by Completing the Square (from file image_ff9541.png)

The Question: Which of the following is/are true, if $x^2 - 8x + 13 = 0$ is solved by completing square method. (Multiple Select Question)

  • 3 should be added on both sides of equation
  • $4 + \sqrt{3}$ is one of the roots
  • $4 - \sqrt{3}$ is one of the roots
  • Equal roots

Core Concept: This question checks both the process of completing the square and the final result (the roots).

Detailed Solution:

  1. Start with the equation: $x^2 - 8x + 13 = 0$

  2. Isolate the x terms: $x^2 - 8x = -13$

  3. Complete the square:

    • The coefficient is $b=-8$.
    • The number to add is $(\frac{-8}{2})^2 = (-4)^2 = 16$.
    • Add 16 to both sides: $x^2 - 8x + 16 = -13 + 16$

  4. Factor the left side and simplify the right: $(x - 4)^2 = 3$

  5. Analyze the first option: The statement says “3 should be added on both sides of equation”. Let’s see if this is a valid step from the start:

    • $x^2 - 8x + 13 = 0$
    • Add 3 to both sides: $x^2 - 8x + 13 + 3 = 0 + 3$
    • $x^2 - 8x + 16 = 3$. This is the same equation we derived in step 4. So, this statement about the process is TRUE.

  6. Solve for the roots from $(x - 4)^2 = 3$:

    • Take the square root of both sides: $x - 4 = \pm\sqrt{3}$
    • Isolate x: $x = 4 \pm\sqrt{3}$
    • The two roots are $x_1 = 4 + \sqrt{3}$ and $x_2 = 4 - \sqrt{3}$.

  7. Evaluate the remaining options:

    • "$4 + \sqrt{3}$ is one of the roots": This is TRUE.
    • "$4 - \sqrt{3}$ is one of the roots": This is TRUE.
    • “Equal roots”: This is FALSE, because the roots $4 + \sqrt{3}$ and $4 - \sqrt{3}$ are distinct.

Final Answer: The true statements are:

  • 3 should be added on both sides of equation
  • $4 + \sqrt{3}$ is one of the roots
  • $4 - \sqrt{3}$ is one of the roots
Solution of quadratic equation using graphSummary Lecture (Quadratic Functions)