Quadratic Equations — Maharashtra (SSC) Class 10 Mathematics Solutions (Free)
Free step-by-step Maharashtra (SSC) Class 10 Mathematics solutions for "Quadratic Equations" — important questions with detailed answers, download PDF for board exam preparation.
TL;DR: Free step-by-step Maharashtra (SSC) Class 10 Mathematics solutions for "Quadratic Equations" — important questions with detailed answers, download PDF…
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Q1: Define a quadratic equation and state the general form.
Quadratic Equation:
A quadratic equation in one variable is a polynomial equation of degree 2.
General form: ax² + bx + c = 0
where a, b, and c are real numbers and a ≠ 0.
The condition a ≠ 0 is essential because if a = 0, it becomes a linear equation, not quadratic.
Examples of quadratic equations:
- x² + 5x + 6 = 0
- 2x² - 7x + 3 = 0
- x² - 9 = 0
- 3x² + 2x = 0
Characteristics:
- The highest power of the variable is 2
- A quadratic equation has at most 2 real roots
- The graph of y = ax² …
Q2: Prove that the discriminant determines the nature of roots of a quadratic equation.
For the quadratic equation ax² + bx + c = 0, the discriminant is Δ = b² - 4ac.
The nature of roots is determined as follows:
1) If Δ > 0 (discriminant is positive):
- The equation has two distinct real roots
- Roots are given by x = (-b ± √Δ)/(2a)
- Example: x² + 5x + 6 = 0 has Δ = 25 - 24 = 1 > 0, so two distinct real roots
2) If Δ = 0 (discriminant is zero):
- The equation has two equal (repeated) real roots
- Both roots are x = -b/(2a)
- Example: x² - 4x + 4 = 0 has…
Q3: Solve x² + 5x + 6 = 0 by factorisation.
Given equation: x² + 5x + 6 = 0
Step 1: Identify factors
We need two numbers that multiply to 6 and add to 5.
Factors of 6 are: 1×6, 2×3
Checking: 2 + 3 = 5 ✓ and 2 × 3 = 6 ✓
Step 2: Factorise
x² + 5x + 6 = (x + 2)(x + 3) = 0
Step 3: Apply zero product property
Either (x + 2) = 0 or (x + 3) = 0
Therefore: x = -2 or x = -3
Step 4: Verification
For x = -2: (-2)² + 5(-2) + 6 = 4 - 10 + 6 = 0 ✓
For x = -3: (-3)² + 5(-3) + 6 = 9 - 15 + 6 = 0 ✓
Therefore, the roots are x = -2 and x = -3
Q4: Solve 2x² - 7x + 3 = 0 using the quadratic formula.
Given equation: 2x² - 7x + 3 = 0
Identify coefficients:
a = 2, b = -7, c = 3
Step 1: Calculate the discriminant
Δ = b² - 4ac
= (-7)² - 4(2)(3)
= 49 - 24
= 25
Since Δ = 25 > 0, the equation has two distinct real roots.
Step 2: Apply the quadratic formula
x = (-b ± √Δ)/(2a)
x = (-(-7) ± √25)/(2 × 2)
x = (7 ± 5)/4
Step 3: Calculate the roots
Root 1: x = (7 + 5)/4 = 12/4 = 3
Root 2: x = (7 - 5)/4 = 2/4 = 1/2
Step 4: Verification
For x = 3: 2(3)² - 7(3) + 3 = 18 - 21 + 3 = 0 ✓
For x = …
Q5: For what values of k does the equation 2x² - 3x + k = 0 have equal roots?
For a quadratic equation to have equal roots, the discriminant must be zero: Δ = 0
Given equation: 2x² - 3x + k = 0
Coefficients: a = 2, b = -3, c = k
Step 1: Write the discriminant condition
Δ = b² - 4ac = 0
(-3)² - 4(2)(k) = 0
9 - 8k = 0
Step 2: Solve for k
8k = 9
k = 9/8
Step 3: Verification
With k = 9/8, the equation becomes:
2x² - 3x + 9/8 = 0
Multiplying by 8: 16x² - 24x + 9 = 0
Discriminant: (-24)² - 4(16)(9) = 576 - 576 = 0 ✓
The equal root is: x = -b/(2a) = -(-3)/(2 × 2) = 3/4
Ve…
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