Q: What are the correct factors of (x^2-6x-27=0)?

((x-9)(x+3)) Since (-9+3=-6) and (-9\times3=-27), ((x-9)(x+3)) is correct. In exams, choose mixed signs carefully.

Q: In which of the following forms can the equation $x^2+18x+81=0$ be written?

$(x+9)^2=0$ Using $(a+b)^2=a^2+2ab+b^2$, we get $(x+9)^2=x^2+18x+81$, since $2\times 9=18$ and $9^2=81$. Therefore, the correct form is $(x+9)^2=0$. The closest distractor, $(x-9)^2$, has the middle term $-18x$, not $+18x$. Exam tip: for a perfect square, compare the square root of the constant term with half the coefficient of $x$.

Q: What are the roots of the equation $4x^2+28x=0$?

$x=0,\,-7$ Factoring the equation gives $4x^2+28x=4x(x+7)=0$. By the zero-product property, $4x=0$ gives $x=0$, and $x+7=0$ gives $x=-7$. Therefore, option A is correct. In option B, the sign of $-7$ is incorrect. In an exam, remember to set both factors equal to zero after factoring.

Q: Which number pair is correct for making the middle term in (x^2+3x-40=0)?

(8) and (-5) (8+(-5)=3) and (8\times(-5)=-40), so this pair is correct. In exams, match the sum with (b) and product with (c).

Question 1

In the completing-square method for x² + 20x + 13 = 0, what number should be added and subtracted?

Accepted Answer

100 The governing rule for completing the square is to add and subtract (b/2)² when the coefficient of x² is 1. In x² + 20x + 13 = 0, the coefficient b is 20. Half of 20 is 10, and squaring it gives 10² = 100. Therefore 100 must be both added and subtracted so that the first three terms can form a perfect square: x² + 20x + 100 = (x + 10)². The equation may then be written as (x + 10)² − 100 + 13 = 0, or (x + 10)² = 87. Hence option A is correct. The number 20 is the linear coefficient, 10 is only its half, and 13 is the constant term; none of these is the required added square.

Question 2

For applying the quadratic formula to 5x² + 2x − 7 = 0, what are a, b, and c?

Accepted Answer

a = 5, b = 2, c = −7 The governing concept is the standard form of a quadratic equation, ax² + bx + c = 0. To identify the coefficients, compare each term directly with that form. In 5x² + 2x − 7 = 0, the coefficient of x² is a = 5, the coefficient of x is b = 2, and the constant term is c = −7. The negative sign belongs to the constant term and must not be omitted. Therefore option A is correct. Option B swaps the coefficients of x² and x. Option C changes both signs of the linear and constant terms, and option D assigns the constant term as a and the other coefficients incorrectly. Correct identification is essential before substituting into the quadratic formula.

Question 3

What will be obtained first after simplifying (10x^2-90=0)?

Accepted Answer

(x^2-9=0) Dividing both sides by (10) gives (x^2-9=0). In exams, simplifying the equation first saves time.

Question 4

What are the correct factors of (x^2-6x-27=0)?

Accepted Answer

((x-9)(x+3)) Since (-9+3=-6) and (-9	imes3=-27), ((x-9)(x+3)) is correct. In exams, choose mixed signs carefully.

Question 5

In which of the following forms can the equation $x^2+18x+81=0$ be written?

Accepted Answer

$(x+9)^2=0$ Using $(a+b)^2=a^2+2ab+b^2$, we get $(x+9)^2=x^2+18x+81$, since $2\times 9=18$ and $9^2=81$. Therefore, the correct form is $(x+9)^2=0$. The closest distractor, $(x-9)^2$, has the middle term $-18x$, not $+18x$. Exam tip: for a perfect square, compare the square root of the constant term with half the coefficient of $x$.

Question 6

What is the repeated root of (x^2+18x+81=0)?

Accepted Answer

(x=-9) ((x+9)^2=0), so the repeated root is (-9). In exams, a perfect square equation has equal roots.

Question 7

By taking out the common factor, how can 4x² + 28x = 0 be written?

Accepted Answer

4x(x + 7) = 0 The governing concept is common-factor extraction, in which the greatest factor shared by every term is placed outside parentheses. Both 4x² and 28x contain 4x. Dividing 4x² by 4x leaves x, while dividing 28x by 4x leaves 7. The plus sign remains plus, so the factorised expression is 4x(x + 7). Expanding it verifies the result: 4x(x + 7) = 4x² + 28x. Therefore option A is correct. Option B loses the factor x and expands to 4x + 28, not the original expression. Options C and D use a minus sign, which would produce a negative 28x term. The factorisation can also lead to solving the equation through the zero-product property.

Question 8

What are the roots of the equation $4x^2+28x=0$?

Accepted Answer

$x=0,\,-7$ Factoring the equation gives $4x^2+28x=4x(x+7)=0$. By the zero-product property, $4x=0$ gives $x=0$, and $x+7=0$ gives $x=-7$. Therefore, option A is correct. In option B, the sign of $-7$ is incorrect. In an exam, remember to set both factors equal to zero after factoring.

Question 9

What will be the roots of (x^2-17x+72=0)?

Accepted Answer

(x=8,9) (x^2-17x+72=(x-8)(x-9)), so the roots are (8) and (9). In exams, check (8+9=17) and (8	imes9=72).

Question 10

Which number pair is correct for making the middle term in (x^2+3x-40=0)?

Accepted Answer

(8) and (-5) (8+(-5)=3) and (8	imes(-5)=-40), so this pair is correct. In exams, match the sum with (b) and product with (c).

Question 11

In the method of splitting the middle term, what is the value of $ac$ for the equation $4x^2+13x+3=0$?

Accepted Answer

12 Comparing the equation with the standard form $ax^2+bx+c=0$, we get $a=4$ and $c=3$. Therefore, $ac=4\times3=12$. The value 13 is $b$, while 3 is only $c$. Exam tip: find $ac$ before splitting the middle term.

Question 12

Which is the correct splitting of the middle term in (4x^2+13x+3=0)?

Accepted Answer

(4x^2+12x+x+3=0) (12+1=13) and (12	imes1=12=ac), so (13x) is split as (12x+x). In exams, keep both sum and product correct.

Question 13

What are the solutions of the equation $x^2-121=0$?

Accepted Answer

$x=\pm 11$ Rewrite $x^2-121=0$ as $x^2-11^2=0$. Then $(x-11)(x+11)=0$, so $x=11$ or $x=-11$, that is, $x=\pm 11$. Option B is incorrect because 121 is the value of $x^2$, not of $x$. In an exam, recognize $121=11^2$ and use the difference-of-squares method.

Question 14

What will be obtained by factoring 9x² − 27x = 0?

Accepted Answer

9x(x − 3) = 0 The governing concept is taking the greatest common factor from every term of a polynomial. In 9x² − 27x, both terms contain 9x. Dividing each term by 9x gives x and −3 respectively, so the expression becomes 9x(x − 3). Therefore the factored equation is 9x(x − 3) = 0, making option A correct. A quick expansion verifies it: 9x multiplied by x gives 9x², and 9x multiplied by −3 gives −27x. Option B incorrectly omits the factor x, option C changes the sign of the constant term, and option D also has the wrong sign. The factor form can then be used with the zero-product property to find x = 0 or x = 3.

Question 15

What are the roots of the equation $9x^2-27x=0$?

Accepted Answer

$x=0, 3$ Factoring the equation gives $9x^2-27x=9x(x-3)=0$. By the zero-product rule, $x=0$ or $x-3=0$, which gives $x=3$. Therefore, the roots are $0$ and $3$. Option B is incorrect because the second root is positive $3$, not $-3$. In an exam, factor out the common term first and then apply the zero-product rule.

Question 16

What is the discriminant $D$ of the equation $x^2+2x+1=0$?

Accepted Answer

0 In the standard form $ax^2+bx+c=0$, $a=1$, $b=2$, and $c=1$. Therefore, $D=b^2-4ac=2^2-4(1)(1)=4-4=0$, so option A is correct. Option D results from taking only $b^2=4$ and not subtracting $4ac$. Exam tip: When $D=0$, the quadratic equation has equal roots.

Question 17

If a quadratic equation has (D=36), what is the correct statement about real roots?

Accepted Answer

Two distinct real roots will be obtained (D=36>0), so two distinct real roots are obtained. In exams, connect (D>0) with distinct real roots.

Question 18

Which is the easiest method to solve x² + 7x + 10 = 0?

Accepted Answer

Factorisation method The governing idea is to select a solution method that matches the structure and coefficients of the quadratic. Here the constant term 10 has factor pairs 1 and 10 or 2 and 5. The pair 5 and 2 has sum 7, so x² + 7x + 10 can be written as (x + 5)(x + 2) = 0. The zero-product property then gives x = −5 or x = −2. Thus factorisation is the easiest and most direct method, making option A correct. Long division is not normally required for solving this quadratic, and a table or graph could be used in some contexts but would be less direct and less exact for this simple expression. The small integer coefficients make factorisation especially convenient.

Question 19

What are the roots of the equation $x^2+7x+10=0$?

Accepted Answer

$x=-5,-2$ The factorisation of $x^2+7x+10$ is $(x+5)(x+2)$. Therefore, $(x+5)(x+2)=0$ gives $x=-5$ or $x=-2$. Option B has the wrong signs; from $x+a=0$, we get $x=-a$. Exam tip: check that the constant terms multiply to 10 and add to 7.

Question 20

While completing the square, which term must be added to (x^2-22x)?

Accepted Answer

(121) Half of (-22) is (-11), and ((-11)^2=121). In exams, the square of half the coefficient is always positive.

Question 21

What are the values of $x$ obtained by solving the equation $x^2=169$ using the square-root method?

Accepted Answer

$x=\pm13$ Taking square roots gives $x=\pm\sqrt{169}=\pm13$, so both $x=13$ and $x=-13$ are solutions. Writing only $x=13$ is incomplete because both the positive and negative numbers have square 169. In an exam, remember to write both values as $x=\pm\sqrt{a}$ when $x^2=a$.

Question 22

What is the correct factorised form of (16x^2-25=0)?

Accepted Answer

((4x-5)(4x+5)=0) Direct answer: Option A, $4x-5)(4x+5)=0$. The expression is a difference of two squares. Write $16x^2=(4x)^2$ and $25=5^2$. The rule is $p^2-q^2=(p-q)(p+q)$, so $16x^2-25=(4x-5)(4x+5)$. Therefore the equation has exactly the factorised form in A. Option A is correct because expansion gives $16x^2+20x-20x-25=16x^2-25$. Option B expands to $16x^2+75x-25$, so it has an unwanted middle term. Option C is a square of one factor and expands to $16x^2-40x+25$, not the given expression. Option D expands to $16x^2-75x-25$, also incorrect. Memory cue: difference of squares means subtract inside the first bracket and add inside the second.

Question 23

What are the roots of (16x^2-25=0)?

Accepted Answer

(x=\frac{5}{4},-\frac{5}{4}) ((4x-5)(4x+5)=0), so (x=\frac{5}{4}) or (x=-\frac{5}{4}). In exams, solve linear factors carefully.

Question 24

Which identity directly applies to (x^2-144=0)?

Accepted Answer

(a^2-b^2=(a-b)(a+b)) The direct answer is option A: a²−b²=(a−b)(a+b). Since 144=12², the expression x²−144 becomes x²−12². This matches a²−b² with a=x and b=12, so it factors as (x−12)(x+12). If solving the equation, (x−12)(x+12)=0 gives x=12 or x=−12. Option A is correct because it represents the difference of two squares. Option B, (a+b)², would expand to a²+2ab+b² and does not match the expression. Option C, (a−b)², would expand to a²−2ab+b² and also has a middle term that is absent here. Option D is not a valid identity: (a+b)² includes 2ab, so a²+b² cannot generally equal it. Memory cue: the pattern square minus square factors into difference times sum.

Question 25

Which are the roots of (x^2+9x+20=0)?

Accepted Answer

(x=-4,-5) (x^2+9x+20=(x+4)(x+5)), so the roots are (-4) and (-5). In exams, a positive middle term can give negative roots.

Methods of Solving Quadratic Equations

Practice questions

01 In the completing-square method for x² + 20x + 13 = 0, what number should be added and subtracted?

02 For applying the quadratic formula to 5x² + 2x − 7 = 0, what are a, b, and c?

03 What will be obtained first after simplifying (10x^2-90=0)?

04 What are the correct factors of (x^2-6x-27=0)?

05 In which of the following forms can the equation \(x^2+18x+81=0\) be written?

06 What is the repeated root of (x^2+18x+81=0)?

07 By taking out the common factor, how can 4x² + 28x = 0 be written?

08 What are the roots of the equation \(4x^2+28x=0\)?

09 What will be the roots of (x^2-17x+72=0)?

10 Which number pair is correct for making the middle term in (x^2+3x-40=0)?

11 In the method of splitting the middle term, what is the value of \(ac\) for the equation \(4x^2+13x+3=0\)?

12 Which is the correct splitting of the middle term in (4x^2+13x+3=0)?

13 What are the solutions of the equation \(x^2-121=0\)?

14 What will be obtained by factoring 9x² − 27x = 0?

15 What are the roots of the equation \(9x^2-27x=0\)?

16 What is the discriminant \(D\) of the equation \(x^2+2x+1=0\)?

17 If a quadratic equation has (D=36), what is the correct statement about real roots?

18 Which is the easiest method to solve x² + 7x + 10 = 0?

19 What are the roots of the equation \(x^2+7x+10=0\)?

20 While completing the square, which term must be added to (x^2-22x)?

21 What are the values of \(x\) obtained by solving the equation \(x^2=169\) using the square-root method?

22 What is the correct factorised form of (16x^2-25=0)?

23 What are the roots of (16x^2-25=0)?

24 Which identity directly applies to (x^2-144=0)?

25 Which are the roots of (x^2+9x+20=0)?

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