Q: Which step is correct in solving (x^2+8x-33=0) by completing square?

((x+4)^2=49) Direct answer: Option A, \$ (x+4)^2=49 \$. Begin with \$x^2+8x-33=0\$, then move the constant: \$x^2+8x=33\$. Half of the coefficient 8 is 4, and \$4^2=16\$. Add 16 to both sides: \$x^2+8x+16=33+16\$. The left side becomes \$ (x+4)^2\$, and the right side becomes 49. Therefore A is correct. B has the wrong sign: \$ (x-4)^2\$ contains \$-8x\$, not +8x. C uses 8 instead of half of 8 and also leaves the right side incorrectly as 33. D uses the correct square form but forgets to add 16 to the right side. Exam cue: for \$x^2+bx\$, add \$ (b/2)^2\$ to both sides.

Q: What are the roots of the equation $x^2+8x-33=0$?

$x=3,-11$ The equation factors as $x^2+8x-33=(x-3)(x+11)=0$. Therefore, $x=3$ or $x=-11$, so option A is correct. In option B, the signs of both roots are incorrect. As an exam check, the sum of the roots should be $-8$ and their product should be $-33$.

Q: What form is obtained by completing the square in (x^2-6x+18=0)?

((x-3)^2+9=0) (x^2-6x+18=(x-3)^2+9), so no real roots are obtained. In exams, use completed square form to understand the nature of roots.

Q: What are the roots obtained by solving the equation \$7x^2=175\$ using the square-root method?

\$x=\pm5\$ Dividing both sides by 7 gives \$x^2=25\$. Therefore, \$x=\pm\sqrt{25}=\pm5\$, so the roots are 5 and -5. Writing only \$x=5\$ or only \$x=-5\$ is incomplete because both square roots must be considered. Exam tip: for \$x^2=a\$, write \$x=\pm\sqrt{a}\$.

Q: Which is the correct solution for (x^2+14x+48=0)?

(x=-6,-8) (x^2+14x+48=(x+6)(x+8)), so (x=-6,-8). In exams, a positive middle term and positive constant can give negative roots.

Question 1

What is the correct way to solve (11x^2=77x)?

Accepted Answer

Write (11x(x-7)=0) From (11x^2-77x=0), (11x(x-7)=0), so (x=0) and (x=7). In exams, dividing by the variable can miss one root.

Question 2

Which step is correct in solving (x^2+8x-33=0) by completing square?

Accepted Answer

((x+4)^2=49) Direct answer: Option A, $ (x+4)^2=49 $. Begin with $x^2+8x-33=0$, then move the constant: $x^2+8x=33$. Half of the coefficient 8 is 4, and $4^2=16$. Add 16 to both sides: $x^2+8x+16=33+16$. The left side becomes $ (x+4)^2$, and the right side becomes 49. Therefore A is correct. B has the wrong sign: $ (x-4)^2$ contains $-8x$, not +8x. C uses 8 instead of half of 8 and also leaves the right side incorrectly as 33. D uses the correct square form but forgets to add 16 to the right side. Exam cue: for $x^2+bx$, add $ (b/2)^2$ to both sides.

Question 3

What are the roots of the equation $x^2+8x-33=0$?

Accepted Answer

$x=3,-11$ The equation factors as $x^2+8x-33=(x-3)(x+11)=0$. Therefore, $x=3$ or $x=-11$, so option A is correct. In option B, the signs of both roots are incorrect. As an exam check, the sum of the roots should be $-8$ and their product should be $-33$.

Question 4

What is the correct splitting of the middle term in (8x^2-2x-3=0)?

Accepted Answer

(8x^2+4x-6x-3=0) (ac=-24) and (4+(-6)=-2), so (-2x) is split as (4x-6x). In exams, check the sign of (ac) carefully.

Question 5

What are the roots of (8x^2-2x-3=0)?

Accepted Answer

(x=\frac{3}{4},-\frac{1}{2}) (8x^2-2x-3=(4x-3)(2x+1)), so the roots are (\frac{3}{4}) and (-\frac{1}{2}). In exams, solve both linear factors carefully.

Question 6

If (D=0) and (a=4,b=-16), what will be the equal root?

Accepted Answer

(x=2) The equal root is (x=\frac{-b}{2a}), so (x=\frac{16}{8}=2). In exams, this short formula is useful when (D=0).

Question 7

What is the correct statement about real roots of (x^2-6x+18=0)?

Accepted Answer

No real roots Here (D=(-6)^2-4(1)(18)=-36<0), so there are no real roots. In exams, (D<0) means no real solution.

Question 8

What form is obtained by completing the square in (x^2-6x+18=0)?

Accepted Answer

((x-3)^2+9=0) (x^2-6x+18=(x-3)^2+9), so no real roots are obtained. In exams, use completed square form to understand the nature of roots.

Question 9

What are the roots obtained by solving the equation $7x^2=175$ using the square-root method?

Accepted Answer

$x=\pm5$ Dividing both sides by 7 gives $x^2=25$. Therefore, $x=\pm\sqrt{25}=\pm5$, so the roots are 5 and -5. Writing only $x=5$ or only $x=-5$ is incomplete because both square roots must be considered. Exam tip: for $x^2=a$, write $x=\pm\sqrt{a}$.

Question 10

Which is the correct solution for (x^2+14x+48=0)?

Accepted Answer

(x=-6,-8) (x^2+14x+48=(x+6)(x+8)), so (x=-6,-8). In exams, a positive middle term and positive constant can give negative roots.

Question 11

Which is the correct factorised form of the equation 11x^2+12x+1=0?

Accepted Answer

(11x+1)(x+1)=0 Expanding (11x+1)(x+1) gives 11x^2+11x+x+1=11x^2+12x+1, so option A is correct. Option C expands to x^2+12x+11, whose coefficient of x^2 is 1 rather than 11. In an exam, multiply the proposed factors to verify the original quadratic equation.

Question 12

What will be the roots of (11x^2+12x+1=0)?

Accepted Answer

(x=-\frac{1}{11},-1) The direct answer is option A: the roots are $x=-\frac{1}{11}$ and $x=-1$. To solve, factor the quadratic. We need two factors whose product is $11x^2+12x+1$. The factorisation is $(11x+1)(x+1)=0$, because multiplying gives $11x^2+11x+x+1=11x^2+12x+1$. By the zero-product rule, either $11x+1=0$, giving $x=-\frac{1}{11}$, or $x+1=0$, giving $x=-1$. Option A is correct because it contains exactly these two values. Option B has positive values, but both roots must be negative because the factors contain plus signs. Option C gives $-11$ instead of $-\frac{1}{11}$; it confuses a coefficient with its reciprocal. Option D has positive values and is therefore incorrect. A useful check is that the sum of roots is $-12/11$, which equals $-1/11-1$, and their product is $1/11$. Memory cue: after factoring, set each factor equal to zero and change the sign carefully.

Question 13

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

Accepted Answer

(x=\frac{-3\pm\sqrt{21}}{2}) Here (D=3^2-4(1)(-3)=21), so (x=\frac{-3\pm\sqrt{21}}{2}). In exams, keep the sign of (c=-3) correct.

Question 14

Which factorised form is correct for (4x^2-12x-7=0)?

Accepted Answer

((2x+1)(2x-7)=0) The correct answer is option A: (2x+1)(2x-7)=0. Expand it carefully: (2x+1)(2x-7)=4x^2-14x+2x-7=4x^2-12x-7, exactly the given quadratic. Therefore option A is correct. Option B expands to (2x-1)(2x+7)=4x^2+14x-2x-7=4x^2+12x-7, so the middle term has the wrong sign. Option C expands to (4x-7)(x+1)=4x^2+4x-7x-7=4x^2-3x-7, not the required middle term. Option D expands to (x+7)(4x-1)=4x^2+28x-x-7=4x^2+27x-7, which is also different. Factoring can always be checked by multiplication. The first terms multiply to 4x^2 and the last terms multiply to -7; the cross terms must combine to -12x. Memory cue: expand every proposed factorisation before accepting it.

Question 15

What are the roots of (4x^2-12x-7=0)?

Accepted Answer

(x=\frac{7}{2},-\frac{1}{2}) ((2x+1)(2x-7)=0), so (x=-\frac{1}{2}) and (\frac{7}{2}). In exams, change signs while writing roots.

Question 16

If a student wrote only (x=9) from (x^2=81), what is the correct correction?

Accepted Answer

One should write (x=\pm9) From (x^2=81), (x=\pm\sqrt{81}=\pm9). In exams, both signs are necessary in the square root method.

Question 17

If (x^2-24x+108=0) is solved by completing the square, which middle step is correct?

Accepted Answer

((x-12)^2=36) Adding (144) to (x^2-24x=-108) gives ((x-12)^2=36). In exams, add the square of half the coefficient to both sides.

Question 18

What are the roots of the equation $x^2-24x+108=0$ when it is solved by completing the square method?

Accepted Answer

$x=6,18$ Writing the equation in completed-square form gives $(x-12)^2=36$. Therefore, $x-12=\pm6$, so $x=12+6=18$ or $x=12-6=6$. Hence, the roots are $6$ and $18$. In an exam, remember to use the $\pm$ sign to obtain both roots; option C contains the correct root $6$ but $12$ is not a root.

Question 19

If (x^2-2kx+25=0) has equal roots and (k>0), what is the value of (k)?

Accepted Answer

(5) For equal roots, (D=0), so ((-2k)^2-4(1)(25)=0) gives (k=5). In exams, apply the condition (k>0).

Question 20

If x² + 4x + 1 = 0, which form is obtained by completing the square?

Accepted Answer

(x + 2)² = 3 Begin by moving the constant term to the right: x² + 4x = −1. To complete the square, add (4/2)² = 2² = 4 to both sides. The left side becomes x² + 4x + 4 = (x+2)², while the right side becomes −1+4=3. Therefore the equivalent equation is (x+2)²=3, so option A is correct. Option B has the wrong sign inside the square and would create −4x. Option C adds an incorrect amount and has the wrong coefficient structure, while option D fails to account for the added 4 on the right. Adding the same quantity to both sides preserves equivalence.

Question 21

What is the root of x² − 2√5x + 5 = 0?

Accepted Answer

x = √5 The governing concept is solving a quadratic equation by recognizing a perfect square. Compare x² − 2√5x + 5 with the identity (x − c)² = x² − 2cx + c². Taking c = √5 gives c² = 5, so the equation becomes x² − 2√5x + 5 = (x − √5)² = 0. A square can equal zero only when its base is zero; hence x − √5 = 0 and x = √5. The root is repeated because both quadratic roots coincide. Therefore option A is correct. Option B has the wrong sign, while C and D confuse the constant 5 with the root and do not satisfy the equation.

Question 22

If x² + 6x + 2 = 0, which form is correct by completing the square?

Accepted Answer

(x + 3)² = 7 The governing concept is completing the square while preserving equality. Start with x² + 6x + 2 = 0 and move the constant term to the other side: x² + 6x = −2. Half the coefficient of x is 6/2 = 3, and its square is 9. Add 9 to both sides, not just one side: x² + 6x + 9 = −2 + 9. The left side becomes (x + 3)² and the right side becomes 7, so (x + 3)² = 7. Therefore option A is correct. Option B has the wrong sign inside the square, option C uses 6 instead of half of 6, and option D adds 9 incorrectly or fails to account for the moved constant.

Question 23

If x^2 + 8x + 5 = 0, which form is correct by completing the square?

Accepted Answer

(x+4)^2 = 11 The governing method is completing the square, which adds the same quantity to both sides so that the quadratic expression becomes a perfect square. Starting with x^2+8x+5=0, move 5 to the right: x^2+8x=-5. Half of the coefficient of x is 8/2=4, and its square is 16. Add 16 to both sides: x^2+8x+16=-5+16=11. The left side factors as (x+4)^2, giving (x+4)^2=11. Thus option A is correct. Option B has the wrong sign because (x-4)^2 produces -8x; option C uses an incorrect square term, and option D omits the required change of 16 on the right-hand side.

Methods of Solving Quadratic Equations

Practice questions

01 What is the correct way to solve (11x^2=77x)?

02 Which step is correct in solving (x^2+8x-33=0) by completing square?

03 What are the roots of the equation \(x^2+8x-33=0\)?

04 What is the correct splitting of the middle term in (8x^2-2x-3=0)?

05 What are the roots of (8x^2-2x-3=0)?

06 If (D=0) and (a=4,b=-16), what will be the equal root?

07 What is the correct statement about real roots of (x^2-6x+18=0)?

08 What form is obtained by completing the square in (x^2-6x+18=0)?

09 What are the roots obtained by solving the equation \\(7x^2=175\\) using the square-root method?

10 Which is the correct solution for (x^2+14x+48=0)?

11 Which is the correct factorised form of the equation 11x^2+12x+1=0?

12 What will be the roots of (11x^2+12x+1=0)?

13 What are the roots of (x^2+3x-3=0) by the quadratic formula?

14 Which factorised form is correct for (4x^2-12x-7=0)?

15 What are the roots of (4x^2-12x-7=0)?

16 If a student wrote only (x=9) from (x^2=81), what is the correct correction?

17 If (x^2-24x+108=0) is solved by completing the square, which middle step is correct?

18 What are the roots of the equation \(x^2-24x+108=0\) when it is solved by completing the square method?

19 If (x^2-2kx+25=0) has equal roots and (k>0), what is the value of (k)?

20 If x² + 4x + 1 = 0, which form is obtained by completing the square?

21 What is the root of x² − 2√5x + 5 = 0?

22 If x² + 6x + 2 = 0, which form is correct by completing the square?

23 If x^2 + 8x + 5 = 0, which form is correct by completing the square?

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