Solution of quadratic equation using Factorization

Question 1: Forming a Quadratic Equation from its Roots (from file image_fffac0.png)

The Question: Choose the correct standard form of a quadratic equation with roots $\frac{7}{2}$ and $-\frac{10}{3}$.

Core Concept: From Roots to Standard Form

If a quadratic equation has roots $r_1$ and $r_2$, it can be written in factored form as:

To get to the standard form ($ax^2 + bx + c = 0$), we expand this expression and clear any fractions.

Detailed Solution:

1. Identify the roots:

   • $r_1 = \frac{7}{2}$
   • $r_2 = -\frac{10}{3}$

2. Write the equation in factored form:

   $$\left(x - \frac{7}{2}\right)\left(x - \left(-\frac{10}{3}\right)\right) = 0$$
   $$\left(x - \frac{7}{2}\right)\left(x + \frac{10}{3}\right) = 0$$

   (Note: This is one of the options, but it is not the standard form with integer coefficients).

3. Clear the fractions inside the parentheses:

   • Multiply the first parenthesis by 2 and the second by 3. To keep the equation balanced, we can think of this as: $$\frac{1}{2}(2x - 7) \cdot \frac{1}{3}(3x + 10) = 0$$ $$\frac{1}{6}(2x - 7)(3x + 10) = 0$$

4. Multiply the entire equation by 6 to eliminate the fraction:

   $$(2x - 7)(3x + 10) = 0$$

5. Expand the expression using the FOIL method (First, Outer, Inner, Last):

   • (First) $2x \cdot 3x = 6x^2$
   • (Outer) $2x \cdot 10 = +20x$
   • (Inner) $-7 \cdot 3x = -21x$
   • (Last) $-7 \cdot 10 = -70$
   • Combine them: $6x^2 + 20x - 21x - 70 = 0$

6. Simplify by combining like terms:

   $$6x^2 - x - 70 = 0$$

Final Answer: The correct standard form is $6x^2 - x - 70 = 0$.

Question 2: Analyzing the Graph of a Parabola (from file image_fffa7b.png)

The Question: Choose the correct option about $a$ with the help of Figure AQ-5.1. (The figure shows an upward-opening parabola with equation $y = a(x + 2)(b - x)$)

Core Concept: Parabola Direction and the Leading Coefficient

The leading coefficient of a quadratic equation determines whether its parabola opens upwards or downwards. For an equation $y = Ax^2 + Bx + C$:

• If $A > 0$ (positive), the parabola opens upwards ($\cup$).
• If $A < 0$ (negative), the parabola opens downwards ($\cap$).

This question is tricky because the equation is in factored form. We must first determine the sign of the $x^2$ term after expansion.

Detailed Solution:

1. Analyze the graph: The parabola in the figure clearly opens upwards. This means its leading coefficient ($A$) must be positive.

2. Analyze the equation: The equation is $y = a(x + 2)(b - x)$.

3. Find the $x^2$ term: To find the leading coefficient, we only need to multiply the terms containing $x$ from each part of the equation:

   • $a \times (x) \times (-x) = a(-x^2) = -ax^2$

4. Relate the equation to the graph:

   • The leading coefficient of the expanded equation is $-a$.
   • Since the parabola opens upwards, this leading coefficient must be positive.
   • Therefore, we have the inequality: $-a > 0$.

5. Solve for $a$:

   • To solve for $a$, we must multiply (or divide) both sides by -1. When you multiply or divide an inequality by a negative number, you must flip the inequality sign.
   • $a < 0$

Final Answer: The correct option is $a < 0$.

Question 3: Finding Roots of Quadratic Functions (from file image_fff71c.png)

The Question: Choose the set of correct options regarding quadratic functions $p(x) = x^2 - 7x + 10$ and $q(x) = x^2 + 7x + 10$. (Multiple Select Question)

Core Concept: Finding Roots by Factoring

To find the roots of a quadratic equation $x^2 + bx + c = 0$, we look for two numbers that multiply to give the constant term ($c$) and add to give the coefficient of the x-term ($b$).

Detailed Solution:

Part 1: Find the roots of $p(x) = x^2 - 7x + 10$

1. We need two numbers that multiply to +10 and add to -7.
2. To get a positive product and a negative sum, both numbers must be negative. The numbers are -5 and -2.
3. The factored form is $(x - 5)(x - 2) = 0$.
4. The roots are $x=5$ and $x=2$.
5. So, the statement “p(x) has two roots 5 and 2” is TRUE.

Part 2: Find the roots of $q(x) = x^2 + 7x + 10$

1. We need two numbers that multiply to +10 and add to +7.
2. The numbers are +5 and +2.
3. The factored form is $(x + 5)(x + 2) = 0$.
4. The roots are $x=-5$ and $x=-2$.
5. So, the statement “q(x) has two roots -5 and -2” is TRUE.

Final Answer: The correct options are:

• $p(x)$ has two roots 5 and 2
• $q(x)$ has two roots -5 and -2

Question 4: Analyzing Roots of a Quadratic Equation (from file image_fff71c.png)

The Question: Identify the incorrect options for the quadratic equation $x^2 - 15x + 54 = 0$. (Multiple Select Question)

• 9 is a root
• Both the roots are equal
• There are no roots
• 6 is a root

Core Concept: Finding and Analyzing Roots

First, find the roots of the equation. Then, evaluate each statement to see if it is true or false. The question asks for the incorrect statements.

Detailed Solution:

1. Find the roots of $x^2 - 15x + 54 = 0$:

   • We need two numbers that multiply to +54 and add to -15.
   • Let’s list factor pairs of 54: (1, 54), (2, 27), (3, 18), (6, 9).
   • The pair (6, 9) adds to 15. To get a sum of -15, both numbers must be negative: -6 and -9.
   • The factored form is $(x - 6)(x - 9) = 0$.
   • The roots are $x=6$ and $x=9$.

2. Evaluate the given options:

   • “9 is a root”: This is TRUE.
   • “Both the roots are equal”: This is FALSE. The roots are 6 and 9, which are distinct.
   • “There are no roots”: This is FALSE. There are two real roots, 6 and 9.
   • “6 is a root”: This is TRUE.

3. Identify the incorrect options:

   • The question asks us to choose the statements that are incorrect. Based on our analysis, two statements are false.

Final Answer: The incorrect options are:

• Both the roots are equal
• There are no roots