Q: Using the square root method, find the solutions of the equation $x^2=64$.

$x=\pm 8$ Taking the square root gives $x=\pm\sqrt{64}=\pm 8$, so the two solutions are $x=8$ and $x=-8$. Writing only $x=8$ or only $x=-8$ is incomplete because both numbers have square 64. In exams, remember to include the $\pm$ sign when taking the square root.

Q: In the completing-the-square method, which number should be added and subtracted to solve the equation $x^2+12x+7=0$?

36 In $x^2+12x+7=0$, the coefficient of $x$ is 12. To form a perfect square, calculate $\left(\frac{12}{2}\right)^2=6^2=36$, so 36 is added and subtracted. Therefore, option A is correct. Exam tip: for $x^2+bx$, use $\left(\frac{b}{2}\right)^2$. Taking 12 or 6 alone does not complete the square.

Q: What are the correct factors of (x^2-4x-21=0)?

((x-7)(x+3)) Since (-7+3=-4) and (-7\times3=-21), ((x-7)(x+3)) is correct. In exams, choose mixed signs carefully.

Q: In which of the following forms can the equation $x^2+14x+49=0$ be written?

$(x+7)^2=0$ In $x^2+14x+49$, we have $49=7^2$ and the middle term is $14x=2\times7\times x$. Therefore, it is the perfect square $(x+7)^2$, so the equation can be written as $(x+7)^2=0$. Option B would produce the middle term $-14x$. In exams, match the expression with $a^2+2ab+b^2=(a+b)^2$.

Q: What is the repeated root of (x^2+14x+49=0)?

$x=-7$ Since $x^2+14x+49=(x+7)^2$, the equation becomes $(x+7)^2=0$, giving $x=-7$. This root occurs twice, so it is the repeated root. Exam tip: for $(x+a)^2=0$, the root is $x=-a$, not $x=a$.

Question 1

Which is the fastest method to solve (x^2-36=0)?

Accepted Answer

Difference of squares method (x^2-36=x^2-6^2), so it is solved quickly by difference of squares. In exams, recognizing (a^2-b^2) saves time.

Question 2

Using the square root method, find the solutions of the equation $x^2=64$.

Accepted Answer

$x=\pm 8$ Taking the square root gives $x=\pm\sqrt{64}=\pm 8$, so the two solutions are $x=8$ and $x=-8$. Writing only $x=8$ or only $x=-8$ is incomplete because both numbers have square 64. In exams, remember to include the $\pm$ sign when taking the square root.

Question 3

Using zero product rule, what are the roots of ((x-7)(x+1)=0)?

Accepted Answer

(x=7,-1) The direct answer is option A: x = 7 or x = −1. The zero-product rule says that if a product is zero, at least one factor must be zero. Thus (x − 7)(x + 1) = 0 gives two separate equations: x − 7 = 0 or x + 1 = 0. Solving them gives x = 7 and x = −1. Option A lists both roots correctly. Option B reverses both signs; substituting x = −7 does not make x − 7 zero, and x = 1 does not make x + 1 zero. Option C has x = 7 correct but changes −1 to 1, which fails the second factor. Option D changes 7 to −7, although x − 7 = 0 clearly requires x = 7. A direct substitution check confirms both answers: at 7, the first factor is zero; at −1, the second factor is zero. Common mistake: do not set both factors to zero simultaneously; use “or,” because either factor being zero makes the product zero.

Question 4

In the completing-the-square method, which number should be added and subtracted to solve the equation $x^2+12x+7=0$?

Accepted Answer

36 In $x^2+12x+7=0$, the coefficient of $x$ is 12. To form a perfect square, calculate $\left(\frac{12}{2}\right)^2=6^2=36$, so 36 is added and subtracted. Therefore, option A is correct. Exam tip: for $x^2+bx$, use $\left(\frac{b}{2}\right)^2$. Taking 12 or 6 alone does not complete the square.

Question 5

For applying the quadratic formula to (4x^2-3x-1=0), what are (a), (b), and (c)?

Accepted Answer

(a=4,b=-3,c=-1) From standard form (ax^2+bx+c=0), (a=4), (b=-3), and (c=-1). In exams, write the signs of (b) and (c) carefully.

Question 6

Which of the following equations is obtained first by simplifying 6x² − 24 = 0?

Accepted Answer

x² − 4 = 0 Dividing every term of 6x² − 24 = 0 by 6 gives x² − 4 = 0, since 24 ÷ 6 = 4. Option D is incorrect because it removes the coefficient from x² but does not divide 24 by 6. In an exam, divide every term of an equation by the same non-zero number.

Question 7

What are the correct factors of (x^2-4x-21=0)?

Accepted Answer

((x-7)(x+3)) Since (-7+3=-4) and (-7	imes3=-21), ((x-7)(x+3)) is correct. In exams, choose mixed signs carefully.

Question 8

In which of the following forms can the equation $x^2+14x+49=0$ be written?

Accepted Answer

$(x+7)^2=0$ In $x^2+14x+49$, we have $49=7^2$ and the middle term is $14x=2\times7\times x$. Therefore, it is the perfect square $(x+7)^2$, so the equation can be written as $(x+7)^2=0$. Option B would produce the middle term $-14x$. In exams, match the expression with $a^2+2ab+b^2=(a+b)^2$.

Question 9

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

Accepted Answer

$x=-7$ Since $x^2+14x+49=(x+7)^2$, the equation becomes $(x+7)^2=0$, giving $x=-7$. This root occurs twice, so it is the repeated root. Exam tip: for $(x+a)^2=0$, the root is $x=-a$, not $x=a$.

Question 10

By taking out the common factor, how can 5x² + 15x = 0 be written?

Accepted Answer

5x(x + 3) = 0 The governing algebraic principle is taking out a common factor from every term while preserving the original expression. Both terms, 5x² and 15x, contain 5x. Dividing the first term by 5x gives x, and dividing the second term by 5x gives 3. Therefore 5x² + 15x = 5x(x + 3), so the equation becomes 5x(x + 3) = 0. Expanding this expression returns 5x² + 15x, confirming the result. Option B omits the common factor x and therefore cannot reproduce the original quadratic term. Option C has an incorrect negative sign, while option D also changes the plus sign to minus. Thus option A is the unique correct form.

Question 11

What are the roots of the equation 5x² + 15x = 0?

Accepted Answer

x = 0, -3 Factoring gives 5x² + 15x = 5x(x + 3) = 0. By the zero-product property, x = 0 or x + 3 = 0, so x = -3. Therefore, the roots are 0 and -3. Option B results from changing the sign of -3 incorrectly. In an exam, remember to include both roots.

Question 12

What will be the roots of (x^2-11x+28=0)?

Accepted Answer

(x=4,7) (x^2-11x+28=(x-4)(x-7)), so the roots are (4) and (7). In exams, check (4+7=11) and (4	imes7=28).

Question 13

Which number pair is correct for making the middle term in (x^2+2x-24=0)?

Accepted Answer

(6) and (-4) (6+(-4)=2) and (6	imes(-4)=-24), so this pair is correct. In exams, match the sum with (b) and product with (c).

Question 14

Using the middle-term splitting method, what is the value of $ac$ for the equation $3x^2+10x+3=0$?

Accepted Answer

9 Comparing the equation with the standard form $ax^2+bx+c=0$, we get $a=3$ and $c=3$. Therefore, $ac=3\times3=9$. The value 10 is $b$, not $ac$. In an exam, calculate $ac$ before splitting the middle term.

Question 15

Which is the correct splitting of the middle term in (3x^2+10x+3=0)?

Accepted Answer

(3x^2+9x+x+3=0) (9+1=10) and (9	imes1=9=ac), so (10x) is split as (9x+x). In exams, keep both sum and product correct.

Question 16

What are the solutions of the equation $x^2-81=0$?

Accepted Answer

$x=\pm 9$ From $x^2-81=0$, we get $x^2=81=9^2$. Hence, $x=9$ or $x=-9$, which is written as $x=\pm9$. Options C and D give only one of the two valid roots, while option B incorrectly treats 81 as the square root of 81. Exam tip: use the difference of squares, $a^2-b^2=(a-b)(a+b)$, to find both roots quickly.

Question 17

What will be obtained by factoring (7x^2-14x=0)?

Accepted Answer

(7x(x-2)=0) Taking common factor (7x) gives (7x(x-2)=0). In exams, you can check by expanding after factoring.

Question 18

What are the roots of the equation $7x^2-14x=0$?

Accepted Answer

$x=0,\;x=2$ Factor the equation as $7x^2-14x=7x(x-2)=0$. By the zero-product property, $7x=0$ or $x-2=0$, giving the roots $x=0$ and $x=2$. In option B, the sign of the second root is incorrect. In an exam, first take out the common factor and then apply the zero-product rule.

Question 19

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

Accepted Answer

0 For a quadratic equation $ax^2+bx+c=0$, the discriminant is $D=b^2-4ac$. Here, $a=1, b=-2, c=1$, so $D=(-2)^2-4(1)(1)=4-4=0$. Therefore, option A is correct. Choosing 4 may result from forgetting to subtract $4ac$. Exam tip: always include the negative sign of $b$ before squaring it.

Question 20

If a quadratic equation has (D=25), what is the correct statement about real roots?

Accepted Answer

Two distinct real roots will be obtained (D=25>0), so two distinct real roots are obtained. In exams, connect (D>0) with distinct real roots.

Question 21

Which is the easiest method for solving x² + 4x + 3 = 0?

Accepted Answer

Factorisation method The governing idea is to select a solution method that matches the structure and coefficients of the quadratic. The expression x² + 4x + 3 has small integer coefficients, and 3 can be factored as 1 × 3 while 1 + 3 = 4. Therefore x² + 4x + 3 = (x + 1)(x + 3). The zero-product property then gives x = −1 or x = −3. Factorisation is direct and efficient here, so option A is correct. A graph could also display the roots, but it is not the easiest exact method. Long division is not the standard method for this task, and “table method” is not an appropriate algebraic procedure.

Question 22

What are the roots of the quadratic equation $x^2+4x+3=0$?

Accepted Answer

$\{-1,-3\}$ Factoring gives $x^2+4x+3=(x+1)(x+3)$. Thus, $(x+1)(x+3)=0$ implies $x=-1$ or $x=-3$, so the roots are $\{-1,-3\}$. Option B has the signs reversed; $x+1=0$ gives $x=-1$. Exam tip: Set each linear factor equal to zero to find the roots.

Question 23

While completing the square, which term must be added to (x^2-18x)?

Accepted Answer

(81) Half of (-18) is (-9), and ((-9)^2=81). In exams, the square of half the coefficient is always positive.

Question 24

What are the values of $x$ when the equation $x^2=121$ is solved by the square root method?

Accepted Answer

$x=\pm 11$ Taking square roots gives $x=\pm\sqrt{121}=\pm 11$, so both $x=11$ and $x=-11$ are solutions. Writing only $x=11$ is incomplete because $(-11)^2$ is also 121; $\pm121$ results from not taking the square root. Exam tip: for $x^2=a$ with $a>0$, always write both roots as $x=\pm\sqrt{a}$.

Question 25

What is the correct factorised form of 9x² − 16 = 0?

Accepted Answer

(3x − 4)(3x + 4) = 0 The governing identity is the difference of two squares: a² − b² = (a − b)(a + b). Rewrite 9x² − 16 as (3x)² − 4². Substituting a = 3x and b = 4 gives (3x − 4)(3x + 4), so the equation becomes (3x − 4)(3x + 4) = 0. Expanding verifies the result: 9x² + 12x − 12x − 16 = 9x² − 16, because the middle terms cancel. Option B expands to an expression containing an incorrect linear term, while option C represents a repeated factor and does not equal the original difference. Option D also gives incorrect coefficients. Hence option A is the correct factorisation.

Methods of Solving Quadratic Equations

Practice questions

01 Which is the fastest method to solve (x^2-36=0)?

02 Using the square root method, find the solutions of the equation \(x^2=64\).

03 Using zero product rule, what are the roots of ((x-7)(x+1)=0)?

04 In the completing-the-square method, which number should be added and subtracted to solve the equation \(x^2+12x+7=0\)?

05 For applying the quadratic formula to (4x^2-3x-1=0), what are (a), (b), and (c)?

06 Which of the following equations is obtained first by simplifying 6x² − 24 = 0?

07 What are the correct factors of (x^2-4x-21=0)?

08 In which of the following forms can the equation \(x^2+14x+49=0\) be written?

09 What is the repeated root of (x^2+14x+49=0)?

10 By taking out the common factor, how can 5x² + 15x = 0 be written?

11 What are the roots of the equation 5x² + 15x = 0?

12 What will be the roots of (x^2-11x+28=0)?

13 Which number pair is correct for making the middle term in (x^2+2x-24=0)?

14 Using the middle-term splitting method, what is the value of \(ac\) for the equation \(3x^2+10x+3=0\)?

15 Which is the correct splitting of the middle term in (3x^2+10x+3=0)?

16 What are the solutions of the equation \(x^2-81=0\)?

17 What will be obtained by factoring (7x^2-14x=0)?

18 What are the roots of the equation \(7x^2-14x=0\)?

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

20 If a quadratic equation has (D=25), what is the correct statement about real roots?

21 Which is the easiest method for solving x² + 4x + 3 = 0?

22 What are the roots of the quadratic equation \(x^2+4x+3=0\)?

23 While completing the square, which term must be added to (x^2-18x)?

24 What are the values of \(x\) when the equation \(x^2=121\) is solved by the square root method?

25 What is the correct factorised form of 9x² − 16 = 0?

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