Quadratic formula

Question 1: Nature of Roots (from file image_a4dfbe.png)

The Question: The quadratic equation $9x^2 + 6x + 1 = 0$ has

• Two distinct roots
• Equal roots
• No roots
• Inadequate information

Core Concept: We can determine the nature of the roots by calculating the discriminant ($\Delta = b^2 - 4ac$).

Detailed Solution:

1. Identify the coefficients from the equation $9x^2 + 6x + 1 = 0$:
   • $a = 9$
   • $b = 6$
   • $c = 1$
2. Calculate the discriminant:
   • $\Delta = b^2 - 4ac$
   • $\Delta = (6)^2 - 4(9)(1)$
   • $\Delta = 36 - 36 = 0$
3. Interpret the result:
   • Since the discriminant is exactly 0, the equation has one repeated real root, which is the same as having “equal roots”.

Final Answer: The equation has Equal roots.

Question 2: Same Discriminant (from file image_a4dfbe.png)

The Question: If two different quadratic equations have same discriminant then choose the correct option.

• The roots of both equation will always be same
• The roots of both equation will always be real
• The roots of both equation will always be distinct
• Discriminant can not determine the relation between the roots of two different equations

Core Concept: The discriminant only determines the nature of the roots (how many and what type), not their actual values. The values of the roots also depend on the ‘$a$’ and ‘$b$’ coefficients from the full quadratic formula.

Detailed Solution:

Let’s test this with an example.

• Equation 1: $x^2 - 4x + 3 = 0$.
  • Discriminant: $\Delta = (-4)^2 - 4(1)(3) = 16 - 12 = 4$.
  • Roots: $(x-1)(x-3)=0 \implies x=1, x=3$.
• Equation 2: $x^2 - 6x + 8 = 0$.
  • Discriminant: $\Delta = (-6)^2 - 4(1)(8) = 36 - 32 = 4$.
  • Roots: $(x-2)(x-4)=0 \implies x=2, x=4$.

Both equations have the same discriminant ($\Delta = 4$), which correctly tells us that both have two distinct real roots. However, the roots themselves (1, 3 vs 2, 4) are completely different.

Therefore, the discriminant only tells you about the type of roots within one equation. It cannot be used to compare the actual root values of two different equations.

Final Answer: Discriminant can not determine the relation between the roots of two different equations.

Question 3: Finding the Roots (from file image_a4dc56.png)

The Question: What will the roots of equation $2x^2 - x - 10 = 0$ be?

Core Concept: When factoring is not immediately obvious, the quadratic formula is a reliable method to find the exact roots of any quadratic equation.

Detailed Solution:

1. Identify the coefficients from the equation $2x^2 - x - 10 = 0$:
   • $a = 2$
   • $b = -1$
   • $c = -10$
2. Apply the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ $$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-10)}}{2(2)}$$
3. Simplify the expression: $$x = \frac{1 \pm \sqrt{1 - (-80)}}{4}$$$$x = \frac{1 \pm \sqrt{1 + 80}}{4}$$$$x = \frac{1 \pm \sqrt{81}}{4}$$ $$x = \frac{1 \pm 9}{4}$$
4. Calculate the two separate roots:
   • Root 1: $x_1 = \frac{1 + 9}{4} = \frac{10}{4} = \frac{5}{2}$
   • Root 2: $x_2 = \frac{1 - 9}{4} = \frac{-8}{4} = -2$

Final Answer: The roots are -2 and 5/2.

Question 4: Identifying Distinct Real Roots (from file image_a4dc56.png)

The Question: Which of the following gives distinct real roots? (Multiple Select Question)

• $69x^2 - 42x - 56 = 0$
• $69x^2 - 10x + 560 = 0$
• $9x^2 - 10x + 1 = 0$
• $9x^2 - 10x - 1 = 0$

Core Concept: An equation has distinct real roots if and only if its discriminant ($\Delta = b^2 - 4ac$) is greater than 0 (positive).

Detailed Solution:

We need to calculate the discriminant for each equation.

• Equation 1: $69x^2 - 42x - 56 = 0$

  • $\Delta = (-42)^2 - 4(69)(-56) = 1764 + 15456 = 17220$.
  • Since $\Delta > 0$, this equation has distinct real roots.

• Equation 2: $69x^2 - 10x + 560 = 0$

  • $\Delta = (-10)^2 - 4(69)(560) = 100 - 154560 = -154460$.
  • Since $\Delta < 0$, this equation has no real roots.

• Equation 3: $9x^2 - 10x + 1 = 0$

  • $\Delta = (-10)^2 - 4(9)(1) = 100 - 36 = 64$.
  • Since $\Delta > 0$, this equation has distinct real roots.

• Equation 4: $9x^2 - 10x - 1 = 0$

  • $\Delta = (-10)^2 - 4(9)(-1) = 100 + 36 = 136$.
  • Since $\Delta > 0$, this equation has distinct real roots.

Final Answer: The equations that give distinct real roots are:

• $69x^2 - 42x - 56 = 0$
• $9x^2 - 10x + 1 = 0$
• $9x^2 - 10x - 1 = 0$