Algebra Chapter 1 Quadratic Equations With One Variable
1. If the factors of a quadratic equation ax2 + bx + c = 0, (a ≠ 0) are (x – α) and (x – β) respectively then ax2 + bx + c = 0 and (x- α)(x- β) are equivalent.
⇒ Here α, β are the roots of that quadratic equation.
2. Sreedhar Acharya’s method to find the roots of ay2 + by + c = 0, (a ≠ 0) is \(x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}\)
3. If α, β are the roots of the quadratic equation ax2 + bx + c = 0, (a ≠ 0) then
⇒ \(\alpha+\beta=-\frac{\text { co-efficient of } x}{\text { co-efficient of } x^2}=-\frac{b}{a}\) and
Read and Learn More WBBSE Solutions for Class 10 Maths
⇒ \(\alpha \beta=\frac{\text { co-efficient of } x^{\circ}}{\text { co-efficient of } x^2}=\frac{c}{a}\)
4. If α, β are the roots of a quadratic equation then the equation will be x2 – (α + β) x + αβ = 0
5. Roots of ax2 + bx + c = 0, (a ≠ 0) are
- Real and unequal if b2 – 4ac > 0
- Real and equal if b2 – 4ac = 0
- Imaginary if b2 – 4ac < 0
6. If one root of a quadratic equation is a + √b (√b is a pure surd) then the other root is the conjugate of a + √b, i.e., a – √b
Class 10 Maths Algebra Chapter 1 Solutions
Roots under particular conditions:
For quadratic equation ax2 + bx + c = 0
1. If b = 0, Roots are real/imaginary as (c < 0 or c > 0) and equal in magnitude but of opposite sign.
2. If c = 0 ⇒ one root is zero, other is \(\frac{b}{a}\).
3. If b = c = 0 ⇒ Both roots are zero.
4. If a = c ⇒ Roots are reciprocal to each other.
5. If a > 0, c < 0; a < 0, c > 0
Roots are of opposite sign.
6. If a > 0, b > 0, c > 0; a < 0, b < 0, c < 0
⇒ Both roots are negative, provided D ≥ 0.
7. If a > 0, b < 0, c > 0; a < 0, b > 0, c < 0
⇒ Both roots are positive, provided D ≤ 0.
8. If sign of a = sign of b ≠ sign of c ⇒ greater root in magnitude is negative.
9. If sign of b = sign of c ≠ sign of a ⇒ greater root in magnitude is positive.
10. If a + b + c = 0 => one root is 1 and the second root is \(\frac{c}{a}\).
Algebra Chapter 1 Quadratic Equations With One Variable True Or False
Example 1. The two roots of the equation x2 – 5x + 6 = 0 are real.
Solution: True
Example 2. The two roots of the equation 2x2 – x – 6 = 0 are not real.
Solution: False
Example 3. One root of the equation ax2 + bx + c = 0, a ≠ 0 is zero when c = 0.
Solution: True
Linear Equations Class 10 Solutions
Example 4. The roots ot the equation ax2 + bx + c = 0 are reciprocal to one another when a = b.
Solution: False
Example 5. If the signs of a and c are opposite to that of b then both roots of the equation ax2 + bx + c = 0, a (≠0) are positive.
Solution: True
Example 6. The roots ot the equation ax2 + bx + c = 0, (a ≠ 0) are equal in magnitude and opposite in signs when c = 0.
Solution: False
Example 7. If b = c = 0, then both roots of the equation ax2 + bx + c = 0, (a ≠ 0) are positive.
Solution: False
Class 10 Algebra Chapter 1 Solved Examples
Example 8. The roots of the equation ax2 + bx + c = 0, (a ≠ 0) are equal when b2 – 4ac > 0.
Solution: False
Example 9. Sum of the roots of 3x2– 5x + 7 = 0 is – 5.
Solution: False
Example 10. Product of the roots of 2x2 – 3x + 7 = 0 is \(\frac{-7}{2}\)
Solution: False
Algebra Chapter 1 Quadratic Equations With One Variable Fill In The Blanks
Example 1. The ratio of the sum and the product of two roots of the equation 7x2– 12x + 18 = 0 _______
Solution: 2: 3
Example 2. If two roots of the equation ax2 + bx + c = 0, (a ≠ 0) are reciprocal to each other, then c = _______
Solution: a
Example 3. If two roots of the equation ax2 + bx + c = 0, (a ≠ 0) are reciprocal to each other and opposite(negative), then a + c = _______
Solution: 0
Example 4. Sum of the roots of 2x2 – 6x + 9 = 0 is _______
Solution: 3
Example 5. If the roots of ax2 – bx + c = 0, (a ≠ 0) are equal then c = _______
Solution: \(\frac{b^2}{4 a}\)
Wbbse Class 10 Algebra Notes
Example 6. If the roots of the equation 3x2 + 8x + 2 = 0 are α, β the \(\frac{1}{\alpha}+\frac{1}{\beta}\) = _____
Solution: -4
Example 7. Roots of (k + 1) x2+ 2kx + (k + 2) = 0, (k ≠ – 1) are equal in magnitude but opposite in sign then k = _____
Solution: 0
Example 8. If the roots of the equation ax2 + bx + c = 0, (a ≠ 0) are reciprocal to each other then c = _______
Solution: a
Example 9. If the product of the roots x2– 3x + k = 0 is -2, then k = _____
Solution: 8
Example 10. Sum of the roots of 3x2 – 5 = 0 is ______
Solution: 0
Linear Equations Formulas Class 10
Algebra Chapter 1 Quadratic Equations With One Variable Short Answer Type Questions
Example 1. One root of equation 3x2 – 5x + c = 0 is 2, find its other root.
Solution: Let the other roots be α
∴ \(\alpha+2=\frac{5}{3} \Rightarrow \alpha=\frac{5}{3}-2=-\frac{1}{3}\)
Example 2. The product of the roots of the equation 3x2+ mn – (2m + 3) = 0 is 5, find m.
Solution: \(-\frac{(2 m+3)}{3}=5\)
⇒ or, – (2m + 3) = 15 or, m = – 9.
Example 3. Form a quadratic equation with rational coefficients where one root is 4 + √7
Solution: Other root is 4 – √7
⇒ Equation is x2 -(4 + √7 +4 – √7)+ (4 + √7)(4- √7) = 0
⇒ or, x2 – 8x + (16 – 7) = 0, or, x2 – 8x + 9 = 0
Example 4. If α, β are the roots of x (x – 3) = 4, find the value of α2 + β2.
Solution: x2 – 3x- 4 = 0
⇒ α + β = 3, αβ = – 4
⇒ α2 + β2 = (α + β)2 – 22β
= 32 -2(-4) = 9 + 8= 17
Class 10 Maths Algebra Important Questions
Example 5. Let us write the quadratic equation if sum of its roots is 14 and product is 24.
Solution: Required equation is x2 – 14x + 24 = 0
Example 6. If the sum and the product of two roots of the equation kx2 + 2x + 3k = 0, (k ≠ 0) are equal, let us write the value of k.
Solution: \(-\frac{2}{k}=\frac{3 k}{k}\) (k ≠ 0)
⇒ or, k = \(\frac{-2}{3}\)
Example 7. If two roots of x2 – 22x + 105 = 0 are α, β, find the value of α – β.
Solution: α + β = 22, αβ = 105
⇒ (α – β)2 = (α + β)2 – 4αβ
= (22)2 – 4 x 105 = 484 – 420 = 64
∴ α – β = ± 8
Example 8. If the sum of two roots of x2 – x = k (2x – 1) is zero, find the value of k.
Solution: x2 – x = 2kx – k
⇒ or, x2 – x(2k + 1) + k = 0
⇒ sum of the roots = 0 or, \(\frac{2 k+1}{1}\) = 0 ⇒ k = \(-\frac{1}{2}\)
Example 9. If one of the roots of the two equations x2 + bx + 12 = 0 and x2 + bx + q = 0 is 2, find the value of q.
Solution: (2)2 + b.2 + 12 = 0
⇒ or, 2b = – 16 ∴ b = -16
⇒ Now, (2)2 + b.2 + q = 0
⇒ or, 4 + (- 8) 2 + q = 0
⇒ or, q = – 4 + 16 = 12
Class 10 Maths Board Exam solutions
Example 10. If one root of x2 – 2x + c = 0 is thrice of another root, then find the value of c.
Solution: Let the roots be α, 3β
⇒ a + 3α = 2
⇒ α, 3α = c
⇒ α = \(\frac{2}{4}\) = \(\frac{1}{2}\)
⇒ or, \(3\left(\frac{1}{2}\right)^2=c\)
⇒ or, c =\(\frac{3}{4}\)