Q: What is the correct factorised form to solve (x^2-2x-15=0)?

((x-5)(x+3)=0) (-5+3=-2) and (-5\times3=-15), so ((x-5)(x+3)) is correct. In exams, match both product and sum.

Q: Using the completing-the-square method, what are the roots of the equation $x^2+6x+1=0$?

$x=-3\pm2\sqrt{2}$ Rewrite the equation as $x^2+6x+9=8$, so $(x+3)^2=8$. Thus, $x+3=\pm\sqrt{8}=\pm2\sqrt{2}$, giving $x=-3\pm2\sqrt{2}$. Option D misses the factor 2 while simplifying $\sqrt{8}$. Exam tip: to complete the square, add the square of half the coefficient of $x$.

Question 1

If x² − 11x = 0 is divided by x and only x − 11 = 0 is written, which root is missed?

Accepted Answer

x = 0 The governing concept is that division by a variable can discard a possible zero value of that variable. First factor the equation without dividing: x² − 11x = x(x − 11), so x(x − 11) = 0. By the zero-product property, either x = 0 or x − 11 = 0, which gives x = 11. If we divide the original equation by x, we implicitly assume x ≠ 0; consequently, the solution x = 0 is lost, and only x = 11 remains in the reduced equation. Therefore option A is correct. Substitution confirms both original roots: 0² − 11(0) = 0 and 11² − 11(11) = 0. The safe method is to factor first and consider every factor separately.

Question 2

What are the roots of (2x^2-7x+3=0) by factorisation method?

Accepted Answer

(x=3,\frac{1}{2}) (2x^2-7x+3=(2x-1)(x-3)), so (x=3) and (x=\frac{1}{2}). In exams, solve each linear factor separately.

Question 3

What is the correct splitting of the middle term in (6x^2+11x+3=0)?

Accepted Answer

(6x^2+9x+2x+3=0) Here (ac=18) and (9+2=11), so (11x) is split as (9x+2x). In exams, check both sum (b) and product (ac).

Question 4

Which are the roots of (3x^2-2x-1=0)?

Accepted Answer

(x=1,-\frac{1}{3}) (3x^2-2x-1=(3x+1)(x-1)), so the roots are (1) and (-\frac{1}{3}). In exams, form mixed-sign factors carefully.

Question 5

What is the correct factorised form to solve (x^2-2x-15=0)?

Accepted Answer

((x-5)(x+3)=0) (-5+3=-2) and (-5	imes3=-15), so ((x-5)(x+3)) is correct. In exams, match both product and sum.

Question 6

Using the quadratic formula, what roots are obtained for (x^2-4x-5=0)?

Accepted Answer

(x=5,-1) Here (D=(-4)^2-4(1)(-5)=36), so (x=\frac{4\pm6}{2}). In exams, do not forget the negative sign of (c) while using the formula.

Question 7

Which step is correct while solving x² + 6x + 1 = 0 by completing the square?

Accepted Answer

(x + 3)² = 8 Completing the square means transforming the quadratic expression into a perfect-square expression without changing the equation. Start with x² + 6x + 1 = 0 and move the constant term: x² + 6x = −1. Half of the coefficient of x is 6/2 = 3, and its square is 9. Add 9 to both sides: x² + 6x + 9 = −1 + 9 = 8. The left side becomes (x + 3)², so (x + 3)² = 8. Therefore option A is correct. Option B uses the full coefficient instead of half, option C has an incorrect right side, and option D has the wrong sign inside the square.

Question 8

Using the completing-the-square method, what are the roots of the equation $x^2+6x+1=0$?

Accepted Answer

$x=-3\pm2\sqrt{2}$ Rewrite the equation as $x^2+6x+9=8$, so $(x+3)^2=8$. Thus, $x+3=\pm\sqrt{8}=\pm2\sqrt{2}$, giving $x=-3\pm2\sqrt{2}$. Option D misses the factor 2 while simplifying $\sqrt{8}$. Exam tip: to complete the square, add the square of half the coefficient of $x$.

Question 9

If (2x^2+5x=3), what are (a,b,c) in standard form (ax^2+bx+c=0)?

Accepted Answer

(a=2,b=5,c=-3) The correct answer is option A: a=2, b=5, c=-3. Standard form means every term must be on the left and the equation must equal zero. Starting with 2x^2+5x=3, subtract 3 from both sides: 2x^2+5x-3=0. Comparing this with ax^2+bx+c=0 gives a=2, b=5, and c=-3. Option A matches all three coefficients. Option B has c=3 because it fails to change the sign when 3 is moved to the left. Option C swaps the coefficients of x^2 and x, so it does not match the equation. Option D changes the signs of the x and constant terms incorrectly. The operation must be applied to both sides, and moving a positive 3 from the right to the left makes it negative. Exam cue: first make the right side zero, then read the coefficients in order.

Question 10

What are the roots of (2x^2+5x-3=0)?

Accepted Answer

(x=\frac{1}{2},-3) (2x^2+5x-3=(2x-1)(x+3)), so (x=\frac{1}{2}) and (-3). In exams, write fractional roots correctly.

Question 11

In which of the following perfect-square forms can the equation $4x^2-12x+9=0$ be written?

Accepted Answer

$(2x-3)^2=0$ Using the identity $(a-b)^2=a^2-2ab+b^2$, we get $4x^2-12x+9=(2x-3)^2$. Hence the equation becomes $(2x-3)^2=0$. In option B, the middle term would be positive, whereas the given middle term is $-12x$. In exams, take the square roots of the first and last terms and then verify the middle term.

Question 12

What is the repeated root of (4x^2-12x+9=0)?

Accepted Answer

(x=\frac{3}{2}) ((2x-3)^2=0), so (2x-3=0) and (x=\frac{3}{2}). In exams, write the repeated root with the correct value.

Question 13

What common mistake can occur while solving (5x^2-20x=0)?

Accepted Answer

Missing (x=0) The correct form is (5x(x-4)=0), giving (x=0) and (x=4). In exams, dividing directly by the variable can miss (x=0).

Question 14

What roots are obtained by solving (3x^2+5x-2=0) by factorisation?

Accepted Answer

(x=\frac{1}{3},-2) (3x^2+5x-2=(3x-1)(x+2)), so the roots are (\frac{1}{3}) and (-2). In exams, solve (3x-1=0) carefully.

Question 15

Using the quadratic formula, what is the value of the discriminant $D$ for the equation $2x^2-4x-3=0$?

Accepted Answer

40 In the standard form $ax^2+bx+c=0$, we have $a=2$, $b=-4$, and $c=-3$. Thus, $D=b^2-4ac=(-4)^2-4(2)(-3)=16+24=40$. Since $c$ is negative, the term $-4ac$ becomes positive. Exam tip: identify the signs of $a$, $b$, and $c$ before substituting; taking only $b^2=16$ is incorrect.

Question 16

What is the correct form of the roots of (2x^2-4x-3=0)?

Accepted Answer

(x=1\pm\frac{\sqrt{10}}{2}) The formula gives (x=\frac{4\pm\sqrt{40}}{4}=1\pm\frac{\sqrt{10}}{2}). In exams, simplify the final form.

Question 17

If x² − 10x + k = 0 has equal roots, what is the value of k?

Accepted Answer

25 For a quadratic equation ax² + bx + c = 0, equal real roots occur precisely when the discriminant D = b² − 4ac is zero. In x² − 10x + k = 0, the coefficients are a = 1, b = −10, and c = k. Substitution gives D = (−10)² − 4(1)(k) = 100 − 4k. Setting the discriminant equal to zero, 100 − 4k = 0, so 4k = 100 and k = 25. Thus option A is correct. Option B is merely the magnitude of the coefficient of x and does not satisfy the discriminant condition. Option C gives D = 20, not zero, while option D incorrectly uses 100 without accounting for the term 4ac. The method also confirms that the repeated root would be 5.

Question 18

If (x^2+kx+16=0) has equal roots and (k>0), what is (k)?

Accepted Answer

(8) For equal roots, (k^2-64=0), so (k=\pm8), and (k>0) gives (k=8). In exams, apply the given condition.

Question 19

Which root is common to (x^2-5x+6=0) and (x^2-6x+8=0)?

Accepted Answer

(x=2) The roots of the first equation are (2,3), and the roots of the second are (2,4). In exams, solve both equations separately and compare roots.

Question 20

For which equation is it not safe to divide by (x) first?

Accepted Answer

(x^2-8x=0) In (x^2-8x=0), (x) is a common factor and (x=0) can be a root. In exams, write (x(x-8)=0) in such cases.

Question 21

In which of the following perfect-square forms can the equation $9x^2-30x+25=0$ be written?

Accepted Answer

$(3x-5)^2=0$ Use the identity $a^2-2ab+b^2=(a-b)^2$. Here, $9x^2=(3x)^2$, $25=5^2$, and the middle term is $-30x=-2\cdot3x\cdot5$. Therefore, $9x^2-30x+25=(3x-5)^2$, so the equation becomes $(3x-5)^2=0$. Option B would produce a positive middle term. In exams, take the square roots of the first and last terms and verify the middle term using $\pm2ab$.

Question 22

What is the root of (9x^2-30x+25=0)?

Accepted Answer

(x=\frac{5}{3}) ((3x-5)^2=0), so (3x-5=0) and (x=\frac{5}{3}). In exams, solve the linear equation after square form.

Question 23

What is obtained when (4x^2-1=8x) is written in standard form?

Accepted Answer

(4x^2-8x-1=0) Bringing (8x) to the left gives (4x^2-8x-1=0). In exams, do not miss any term while making standard form.

Question 24

What is the discriminant $D$ of the equation $4x^2-8x-1=0$?

Accepted Answer

80 For a quadratic equation $ax^2+bx+c=0$, the discriminant is $D=b^2-4ac$. Here, $a=4$, $b=-8$, and $c=-1$, so $D=(-8)^2-4(4)(-1)=64+16=80$. The value 48 results from incorrectly treating the contribution of $-4ac$ as negative. Exam tip: when $c$ is negative, $-4ac$ contributes a positive value.

Question 25

Which middle step is correct while solving (x^2-12x+20=0) by completing square?

Accepted Answer

((x-6)^2=16) From (x^2-12x+20=0), (x^2-12x=-20), then adding (36) gives ((x-6)^2=16). In exams, add the same number to both sides.

Methods of Solving Quadratic Equations

Practice questions

01 If x² − 11x = 0 is divided by x and only x − 11 = 0 is written, which root is missed?

02 What are the roots of (2x^2-7x+3=0) by factorisation method?

03 What is the correct splitting of the middle term in (6x^2+11x+3=0)?

04 Which are the roots of (3x^2-2x-1=0)?

05 What is the correct factorised form to solve (x^2-2x-15=0)?

06 Using the quadratic formula, what roots are obtained for (x^2-4x-5=0)?

07 Which step is correct while solving x² + 6x + 1 = 0 by completing the square?

08 Using the completing-the-square method, what are the roots of the equation \(x^2+6x+1=0\)?

09 If (2x^2+5x=3), what are (a,b,c) in standard form (ax^2+bx+c=0)?

10 What are the roots of (2x^2+5x-3=0)?

11 In which of the following perfect-square forms can the equation \(4x^2-12x+9=0\) be written?

12 What is the repeated root of (4x^2-12x+9=0)?

13 What common mistake can occur while solving (5x^2-20x=0)?

14 What roots are obtained by solving (3x^2+5x-2=0) by factorisation?

15 Using the quadratic formula, what is the value of the discriminant \(D\) for the equation \(2x^2-4x-3=0\)?

16 What is the correct form of the roots of (2x^2-4x-3=0)?

17 If x² − 10x + k = 0 has equal roots, what is the value of k?

18 If (x^2+kx+16=0) has equal roots and (k>0), what is (k)?

19 Which root is common to (x^2-5x+6=0) and (x^2-6x+8=0)?

20 For which equation is it not safe to divide by (x) first?

21 In which of the following perfect-square forms can the equation \(9x^2-30x+25=0\) be written?

22 What is the root of (9x^2-30x+25=0)?

23 What is obtained when (4x^2-1=8x) is written in standard form?

24 What is the discriminant \(D\) of the equation \(4x^2-8x-1=0\)?

25 Which middle step is correct while solving (x^2-12x+20=0) by completing square?

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