Q: Which middle step is correct while solving (x^2-10x+24=0) by completing the square?

((x-5)^2=1) Direct answer: Option A, \$ (x-5)^2=1 \$. Start with \$x^2-10x+24=0\$. Move the constant term: \$x^2-10x=-24\$. Half the coefficient of x is \$-10/2=-5\$, and its square is 25. Add 25 to both sides: \$x^2-10x+25=-24+25\$. The left side is \$ (x-5)^2\$, and the right side is 1. Thus A is correct. B has the wrong sign and would represent \$x^2+10x+25\$. C wrongly uses \$x-10\$ and ignores the half-coefficient rule. D leaves the right side as 24 and forgets the added 25. Option A is the exact middle step; later, it gives \$x=4\$ or \$x=6\$. Memory cue: add the square of half the x-coefficient to both sides.

Q: What is the value of the discriminant $D$ used in the quadratic formula for the equation $5x^2-10x-3=0$?

160 Comparing the equation with $ax^2+bx+c=0$, we get $a=5$, $b=-10$, and $c=-3$. Thus, $D=b^2-4ac=(-10)^2-4(5)(-3)=100+60=160$. Since $c$ is negative, the term $-4ac$ becomes positive. Exam tip: write the sign of $b$ explicitly before squaring it.

Question 1

Which middle step is correct while solving (x^2-10x+24=0) by completing the square?

Accepted Answer

((x-5)^2=1) Direct answer: Option A, $ (x-5)^2=1 $. Start with $x^2-10x+24=0$. Move the constant term: $x^2-10x=-24$. Half the coefficient of x is $-10/2=-5$, and its square is 25. Add 25 to both sides: $x^2-10x+25=-24+25$. The left side is $ (x-5)^2$, and the right side is 1. Thus A is correct. B has the wrong sign and would represent $x^2+10x+25$. C wrongly uses $x-10$ and ignores the half-coefficient rule. D leaves the right side as 24 and forgets the added 25. Option A is the exact middle step; later, it gives $x=4$ or $x=6$. Memory cue: add the square of half the x-coefficient to both sides.

Question 2

What roots are obtained by solving (12x^2+17x+5=0) by factorisation?

Accepted Answer

(x=-1,-\frac{5}{12}) (12x^2+17x+5=(12x+5)(x+1)), so the roots are (-\frac{5}{12}) and (-1). In exams, positive factors give negative roots.

Question 3

What is the correct splitting of the middle term in (4x^2-19x-5=0)?

Accepted Answer

(4x^2-20x+x-5=0) Here (ac=-20) and (-20+1=-19), so the middle term is (-20x+x). In exams, check the sign of (ac) carefully.

Question 4

What are the roots of (4x^2-19x-5=0)?

Accepted Answer

(x=5,-\frac{1}{4}) (4x^2-19x-5=(4x+1)(x-5)), so the roots are (5) and (-\frac{1}{4}). In exams, reverse the signs from linear factors to write roots.

Question 5

What are the roots of (5x^2+9x-2=0)?

Accepted Answer

(x=\frac{1}{5},-2) (5x^2+9x-2=(5x-1)(x+2)), so the roots are (\frac{1}{5}) and (-2). In exams, correct standard form is very important.

Question 6

What is the repeated root of (36x^2-60x+25=0)?

Accepted Answer

(x=\frac{5}{6}) ((6x-5)^2=0), so (6x-5=0) and (x=\frac{5}{6}). In exams, write the repeated root as a correct fraction.

Question 7

Which mistake should be avoided while solving (8x^2-32x=0)?

Accepted Answer

Missing (x=0) (8x^2-32x=8x(x-4)), so (x=0) and (x=4) are both roots. In exams, dividing by the variable can miss (x=0).

Question 8

What is the value of the discriminant $D$ used in the quadratic formula for the equation $5x^2-10x-3=0$?

Accepted Answer

160 Comparing the equation with $ax^2+bx+c=0$, we get $a=5$, $b=-10$, and $c=-3$. Thus, $D=b^2-4ac=(-10)^2-4(5)(-3)=100+60=160$. Since $c$ is negative, the term $-4ac$ becomes positive. Exam tip: write the sign of $b$ explicitly before squaring it.

Question 9

What is the correct form of the roots of (5x^2-10x-3=0)?

Accepted Answer

(x=1\pm\frac{2\sqrt{10}}{5}) The formula gives (x=\frac{10\pm\sqrt{160}}{10}=1\pm\frac{2\sqrt{10}}{5}). In exams, simplify (\sqrt{160}=4\sqrt{10}).

Question 10

If x² − 16x + k = 0 has equal roots, what is the value of k?

Accepted Answer

64 The governing concept is the discriminant condition for equal roots. A quadratic equation ax² + bx + c = 0 has equal real roots precisely when its discriminant D = b² − 4ac is zero. In x² − 16x + k = 0, the coefficients are a = 1, b = −16, and c = k. Therefore D = (−16)² − 4(1)(k) = 256 − 4k. Setting D equal to zero gives 256 − 4k = 0, so 4k = 256 and k = 64. Hence option A is correct. The value 16 is only the magnitude of the linear coefficient, not k. The value 32 reflects incomplete division, while 256 ignores the factor 4ac. With k = 64, the equation becomes (x − 8)² = 0, confirming equal roots.

Question 11

If (x^2+kx+36=0) has equal roots and (k>0), what is (k)?

Accepted Answer

(12) For equal roots, (k^2-144=0), so (k=\pm12), and (k>0) gives (k=12). In exams, apply the given condition.

Question 12

Which root is common to (x^2-11x+30=0) and (x^2-13x+42=0)?

Accepted Answer

(x=6) The roots of the first equation are (5,6), and the roots of the second are (6,7). In exams, solve both equations separately and compare the common root.

Question 13

In which of the following perfect-square forms can the equation $49x^2-42x+9=0$ be written?

Accepted Answer

$(7x-3)^2=0$ Using $(a-b)^2=a^2-2ab+b^2$, we get $(7x-3)^2=49x^2-42x+9$. Hence the equation becomes $(7x-3)^2=0$. In option B, the middle term would be $+42x$, while options C and D have an incorrect coefficient of $x^2$. Exam tip: take the square roots of the first and last terms and verify the middle term using $2ab$.

Question 14

What is the root of (49x^2-42x+9=0)?

Accepted Answer

(x=\frac{3}{7}) ((7x-3)^2=0), so (7x-3=0) and (x=\frac{3}{7}). In exams, solve the linear equation after square form.

Question 15

What is obtained when the equation $3x^2+2=11x$ is written in the standard form $ax^2+bx+c=0$?

Accepted Answer

$3x^2-11x+2=0$ In standard form, all terms are brought to one side and the other side is made zero. Subtracting $11x$ from both sides gives $3x^2+2-11x=0$, which can be written as $3x^2-11x+2=0$. Therefore, option A is correct. Option B has the wrong sign for the linear term. Exam tip: a term changes its sign when it is transposed to the other side.

Question 16

What is the discriminant $D$ of the quadratic equation $3x^2-11x+2=0$?

Accepted Answer

97 In the standard form $ax^2+bx+c=0$, we have $a=3$, $b=-11$, and $c=2$. Thus, the discriminant is $D=b^2-4ac=(-11)^2-4(3)(2)=121-24=97$. The value 121 results from failing to subtract $4ac$. In an exam, identify $a,b,c$ first and then apply $D=b^2-4ac$.

Question 17

Which middle step is correct while solving (x^2-18x+45=0) by completing square?

Accepted Answer

((x-9)^2=36) The correct answer is option A: (x-9)^2=36. Start with x^2-18x+45=0 and move 45 to the right: x^2-18x=-45. Half of the coefficient of x is -18/2=-9, and its square is 81. Add 81 to both sides: x^2-18x+81=-45+81=36. The left side becomes (x-9)^2, so the required middle step is (x-9)^2=36. Option A is correct. Option B has (x+9)^2, whose middle term would be +18x, not -18x. Option C uses 18 instead of half of 18 and does not complete the square. Option D leaves the right side as 45, although the original 45 was moved and then must be adjusted after adding 81. Memory cue: half the x-coefficient, square it, and add it to both sides.

Question 18

What are the roots of the quadratic equation $x^2-18x+45=0$?

Accepted Answer

$x=3,15$ The equation factors as $(x-3)(x-15)=0$. Therefore, $x=3$ or $x=15$, so option A is correct. In option C, the numbers add to 18 but their product is 72, whereas the required product is 45. In an exam, verify the roots using their sum 18 and product 45.

Question 19

If the roots of the equation $x^2+px+42=0$ are $-6$ and $-7$, what is the value of $p$?

Accepted Answer

13 The given roots form the factors $(x+6)(x+7)=0$. Expanding gives $x^2+13x+42=0$, so $p=13$. The distractor $-13$ results from missing the negative sign in the relation that the sum of roots is $-p$: $-6+(-7)=-13=-p$. Exam tip: For $x^2+px+q=0$, the sum of the roots is $-p$.

Question 20

If (4) and (9) are roots of a quadratic equation, which equation can it be?

Accepted Answer

(x^2-13x+36=0) If roots are (4) and (9), then ((x-4)(x-9)=0), that is (x^2-13x+36=0). In exams, form factors with opposite signs of roots.

Question 21

What will be the roots after solving (5x^2+16x+3=0)?

Accepted Answer

(x=-3,-\frac{1}{5}) (5x^2+16x+3=(5x+1)(x+3)), so the roots are (-\frac{1}{5}) and (-3). In exams, positive factors give negative roots.

Question 22

Which statement is correct for (x^2-4x-32=0)?

Accepted Answer

The roots are (8) and (-4) (x^2-4x-32=(x-8)(x+4)), so the roots are (8) and (-4). In exams, the larger value decides the sign of the middle term.

Question 23

What roots are obtained by putting (a=1,b=-14,c=45) in the quadratic formula?

Accepted Answer

(x=5,9) (D=(-14)^2-4(1)(45)=16), so (x=\frac{14\pm4}{2}) gives (5) and (9). In exams, keep the sign of (b) correct in the formula.

Question 24

Which number pair helps in making the middle term in (x^2-20x+96=0)?

Accepted Answer

(-8) and (-12) (-8+(-12)=-20) and ((-8)(-12)=96), so this pair is correct. In exams, when (c) is positive and (b) is negative, both numbers are negative.

Question 25

What are the roots of (x^2-20x+96=0)?

Accepted Answer

(x=8,12) (x^2-20x+96=(x-8)(x-12)), so the roots are (8) and (12). In exams, change signs while writing roots from factors.

Methods of Solving Quadratic Equations

Practice questions

01 Which middle step is correct while solving (x^2-10x+24=0) by completing the square?

02 What roots are obtained by solving (12x^2+17x+5=0) by factorisation?

03 What is the correct splitting of the middle term in (4x^2-19x-5=0)?

04 What are the roots of (4x^2-19x-5=0)?

05 What are the roots of (5x^2+9x-2=0)?

06 What is the repeated root of (36x^2-60x+25=0)?

07 Which mistake should be avoided while solving (8x^2-32x=0)?

08 What is the value of the discriminant \(D\) used in the quadratic formula for the equation \(5x^2-10x-3=0\)?

09 What is the correct form of the roots of (5x^2-10x-3=0)?

10 If x² − 16x + k = 0 has equal roots, what is the value of k?

11 If (x^2+kx+36=0) has equal roots and (k>0), what is (k)?

12 Which root is common to (x^2-11x+30=0) and (x^2-13x+42=0)?

13 In which of the following perfect-square forms can the equation \(49x^2-42x+9=0\) be written?

14 What is the root of (49x^2-42x+9=0)?

15 What is obtained when the equation \(3x^2+2=11x\) is written in the standard form \(ax^2+bx+c=0\)?

16 What is the discriminant \(D\) of the quadratic equation \(3x^2-11x+2=0\)?

17 Which middle step is correct while solving (x^2-18x+45=0) by completing square?

18 What are the roots of the quadratic equation \(x^2-18x+45=0\)?

19 If the roots of the equation \(x^2+px+42=0\) are \(-6\) and \(-7\), what is the value of \(p\)?

20 If (4) and (9) are roots of a quadratic equation, which equation can it be?

21 What will be the roots after solving (5x^2+16x+3=0)?

22 Which statement is correct for (x^2-4x-32=0)?

23 What roots are obtained by putting (a=1,b=-14,c=45) in the quadratic formula?

24 Which number pair helps in making the middle term in (x^2-20x+96=0)?

25 What are the roots of (x^2-20x+96=0)?

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