Q: What will be the factors of (x^2-19x+90=0)?

((x-9)(x-10)) (9+10=19) and (9\times10=90), so ((x-9)(x-10)) is correct. In exams, take both negative signs for a negative middle term.

Q: Which is the correct factorised form of the equation $3x^2-11x+6=0$?

$(3x-2)(x-3)=0$ Expanding $(3x-2)(x-3)$ gives $3x^2-9x-2x+6=3x^2-11x+6$, so option A is correct. Option C expands to $3x^2-9x+6$, whose coefficient of $x$ is incorrect. In an exam, expand the factors to check both the middle term and the constant term.

Q: What are the roots of the equation \$3x^2-11x+6=0\$?

\$x=3,\frac{2}{3}\$ Factoring the quadratic gives \$3x^2-11x+6=(3x-2)(x-3)\$. Therefore, \$3x-2=0\$ gives \$x=\frac{2}{3}\$, and \$x-3=0\$ gives \$x=3\$. Hence, option A is correct. Option D incorrectly uses \$\frac{3}{2}\$ instead of \$\frac{2}{3}\$. In an exam, set each factor equal to zero and solve for the roots separately.

Q: Which values of \$x\$ are obtained by solving \$x-4)^2=25\$?

\$x=9,-1\$ Taking square roots gives \$x-4=\\pm 5\$. Thus, \$x-4=5\$ gives \$x=9\$, while \$x-4=-5\$ gives \$x=-1\$. Option B results from handling the addition of 4 incorrectly. In exams, remember to include both \$\\pm\$ cases when taking a square root.

Q: While solving $x^2+24x+17=0$ by the completing-square method, which number is added to make $x^2+24x$ a perfect square?

144 The coefficient of $x$ is 24, so its half is 12. To complete the square, we add the square of this half: $\left(\frac{24}{2}\right)^2=12^2=144$. Therefore, 144 is added to both sides. Remember that 12 is the half of the coefficient, while 144 is the number added.

Q: What are the correct factors of (x^2+6x-55=0)?

((x+11)(x-5)) (11+(-5)=6) and (11\times(-5)=-55), so ((x+11)(x-5)) is correct. In exams, keep one sign positive and one negative.

Question 1

Why is (x=0) a root of (x^2-11x=0)?

Accepted Answer

Because (x(x-11)=0) (x^2-11x=x(x-11)), so zero product rule gives (x=0). In exams, do not lose this root by dividing by the variable.

Question 2

What will be the factors of (x^2-19x+90=0)?

Accepted Answer

((x-9)(x-10)) (9+10=19) and (9	imes10=90), so ((x-9)(x-10)) is correct. In exams, take both negative signs for a negative middle term.

Question 3

What are the roots of (x^2-19x+90=0)?

Accepted Answer

(x=9,10) ((x-9)(x-10)=0), so (x=9) and (x=10). In exams, roots are obtained by taking opposite signs of factors.

Question 4

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

Accepted Answer

(3x^2-9x-2x+6=0) ((-9)+(-2)=-11) and ((-9)(-2)=18=ac), so (-9x-2x) is correct. In exams, split the middle term by checking (ac).

Question 5

Which is the correct factorised form of the equation $3x^2-11x+6=0$?

Accepted Answer

$(3x-2)(x-3)=0$ Expanding $(3x-2)(x-3)$ gives $3x^2-9x-2x+6=3x^2-11x+6$, so option A is correct. Option C expands to $3x^2-9x+6$, whose coefficient of $x$ is incorrect. In an exam, expand the factors to check both the middle term and the constant term.

Question 6

What are the roots of the equation $3x^2-11x+6=0$?

Accepted Answer

$x=3,\frac{2}{3}$ Factoring the quadratic gives $3x^2-11x+6=(3x-2)(x-3)$. Therefore, $3x-2=0$ gives $x=\frac{2}{3}$, and $x-3=0$ gives $x=3$. Hence, option A is correct. Option D incorrectly uses $\frac{3}{2}$ instead of $\frac{2}{3}$. In an exam, set each factor equal to zero and solve for the roots separately.

Question 7

Which values of $x$ are obtained by solving $x-4)^2=25$?

Accepted Answer

$x=9,-1$ Taking square roots gives $x-4=\pm 5$. Thus, $x-4=5$ gives $x=9$, while $x-4=-5$ gives $x=-1$. Option B results from handling the addition of 4 incorrectly. In exams, remember to include both $\pm$ cases when taking a square root.

Question 8

While solving $x^2+24x+17=0$ by the completing-square method, which number is added to make $x^2+24x$ a perfect square?

Accepted Answer

144 The coefficient of $x$ is 24, so its half is 12. To complete the square, we add the square of this half: $\left(\frac{24}{2}\right)^2=12^2=144$. Therefore, 144 is added to both sides. Remember that 12 is the half of the coefficient, while 144 is the number added.

Question 9

Which of the following is equal to x² + 24x + 144?

Accepted Answer

(x + 12)² Using (a + b)² = a² + 2ab + b², we get (x + 12)² = x² + 2(x)(12) + 12² = x² + 24x + 144. Therefore, option A is correct. The minus sign in (x − 12)² would instead produce a middle term of −24x.

Question 10

What are the correct factors of (x^2+6x-55=0)?

Accepted Answer

((x+11)(x-5)) (11+(-5)=6) and (11	imes(-5)=-55), so ((x+11)(x-5)) is correct. In exams, keep one sign positive and one negative.

Question 11

What are the roots of (x^2+6x-55=0)?

Accepted Answer

(x=5,-11) ((x+11)(x-5)=0), so (x=-11) and (x=5). In exams, the sign changes while finding roots from factors.

Question 12

What is the correct simplified form of the equation $12x^2=108$ for solving it?

Accepted Answer

$x^2=9$ Dividing both sides of the equation by 12 gives $\frac{12x^2}{12}=\frac{108}{12}$, so $x^2=9$. Solving further gives $x=\pm3$. In an exam, remember to perform the same operation on both sides of an equation.

Question 13

What are the solutions of the equation $12x^2=108$?

Accepted Answer

$x=\pm3$ Dividing both sides by 12 gives $x^2=9$. Therefore, $x=\pm\sqrt{9}=\pm3$, so the two solutions are $x=3$ and $x=-3$. Writing only $x=3$ is incomplete because the square of $-3$ is also 9. In exams, remember to include both the positive and negative square roots.

Question 14

What is the correct part inside the square root in the quadratic formula?

Accepted Answer

(b^2-4ac) In the quadratic formula, the part inside the square root is (b^2-4ac). In exams, it is also called the discriminant (D).

Question 15

If $x^2+10x+k$ is a perfect square and can be written as $(x+5)^2$, what is the value of $k$?

Accepted Answer

25 $(x+5)^2=x^2+2(5)x+5^2=x^2+10x+25$. Comparing this with $x^2+10x+k$ gives $k=25$. The distractor 5 is the constant inside the bracket, whereas $k$ is its square. Exam tip: in $(x+a)^2=x^2+2ax+a^2$, the constant term is $a^2$.

Question 16

In which method is the equation first changed into a form like ((x+p)^2=q)?

Accepted Answer

Completing square method In completing square method, the quadratic part is made into the form ((x+p)^2). In exams, this method is useful when simple factors are not found quickly.

Question 17

What is the correct factorised form of (25x^2-1=0)?

Accepted Answer

((5x-1)(5x+1)=0) (25x^2-1=(5x)^2-1^2), so ((5x-1)(5x+1)=0) is correct. In exams, quickly identify the difference of squares.

Question 18

What are the roots of 5x² + 6x + 1 = 0?

Accepted Answer

x = −1/5, −1 The equation is most efficiently solved by factorisation. We look for two binomials whose product has first term 5x², constant term 1, and middle term 6x. The factorisation is 5x² + 6x + 1 = (5x + 1)(x + 1), because multiplication gives 5x² + 5x + x + 1 = 5x² + 6x + 1. By the zero-product property, either 5x + 1 = 0 or x + 1 = 0. These equations give x = −1/5 and x = −1, respectively. Therefore option A is correct. Option B changes both signs, option C treats coefficients as roots rather than solving the factors, and option D gives positive values that do not satisfy the original equation. Direct substitution confirms both negative roots.

Question 19

In which perfect-square form can 16x² − 24x + 9 = 0 be written?

Accepted Answer

(4x − 3)² = 0 The governing algebraic identity is (p−q)²=p²−2pq+q². In the given quadratic, 16x² is (4x)² and the constant term 9 is 3². Taking p=4x and q=3 gives (4x−3)²=(4x)²−2(4x)(3)+3²=16x²−24x+9. Hence the equation is exactly (4x−3)²=0, so option A is correct. Expanding option B would produce a positive middle term, +24x, rather than −24x. Option C has leading term 256x², and option D has leading term x², so neither can represent the original expression. The identity also shows that the repeated root would be x=3/4, although finding the root is not needed to identify the form.

Question 20

If 5x² + 9x = 2, what is the equation in standard form?

Accepted Answer

5x² + 9x − 2 = 0 A quadratic equation is commonly written in standard form as ax² + bx + c = 0, with a ≠ 0. Starting from 5x² + 9x = 2, subtract 2 from both sides to preserve equality: 5x² + 9x − 2 = 2 − 2 = 0. Thus the standard-form equation is 5x² + 9x − 2 = 0, which is option A. Option B results from changing the sign incorrectly while moving 2. Option C changes the sign of the 9x term without justification, and option D interchanges the coefficients of x² and x. No solving of the quadratic is required; the task only asks for rearrangement into standard form.

Question 21

In which perfect-square form can 36x² − 60x + 25 = 0 be written?

Accepted Answer

(6x − 5)² = 0 The governing concept is the identity (p − q)² = p² − 2pq + q². Choose p = 6x and q = 5. Then (6x − 5)² = (6x)² − 2(6x)(5) + 5² = 36x² − 60x + 25. Thus the given quadratic equation is exactly equivalent to (6x − 5)² = 0, so option A is correct. The plus form in option B would produce a positive middle term, +60x. In option C, the leading term would be (36x)² = 1296x², not 36x². In option D, the leading term would be x². Matching the leading, middle, and constant coefficients confirms that only option A represents the given perfect square.

Question 22

On solving (x − 7)² = 11, what are the possible values of x?

Accepted Answer

x = 7 ± √11 To solve an equation containing a square, take both square roots because a real number can have either a positive or a negative square root. From (x−7)²=11, we obtain x−7=±√11. Adding 7 to both sides gives x=7±√11. Thus the two solutions are x=7+√11 and x=7−√11, making option A correct. Option B incorrectly changes the sign of 7 when it is transposed. Option C uses 11 instead of its square root, and option D places the square root on the wrong number and also omits the correct structure. The ± symbol is essential because both values square to 11.

Question 23

If ((x-3)(x-7)=0), what are the solutions by zero product rule?

Accepted Answer

x=3 or x=7 The governing concept is the zero-product rule: if the product of two real expressions is zero, at least one factor must be zero. Therefore, from (x−3)(x−7)=0, set the factors separately equal to zero: x−3=0 gives x=3, and x−7=0 gives x=7. Thus the solution set is {3,7}, so option A is correct. Option B incorrectly changes both signs, while options C and D do not make either original factor zero in the required alternatives. Substitution confirms both answers: for x=3 the first factor is zero, and for x=7 the second factor is zero.

Question 24

If (x-4)(x-9)=0, what are the solutions by the zero-product rule?

Accepted Answer

x = 4, 9 The governing principle is the zero-product rule: if the product of two factors is zero, at least one factor must be zero. Applying it to (x-4)(x-9)=0 gives two cases: x-4=0 or x-9=0. Solving them separately gives x=4 and x=9, so the solution set is {4,9}. Therefore option A is correct. Option B changes the signs incorrectly; substituting -4 or -9 does not make the corresponding factors zero. Option C has no valid connection with the factorized equation, and option D incorrectly treats the constants as though their sum and product were the roots. Since the expression is already factorized, the zero-product rule gives the solutions directly and no quadratic formula is needed.

Methods of Solving Quadratic Equations

Practice questions

01 Why is (x=0) a root of (x^2-11x=0)?

02 What will be the factors of (x^2-19x+90=0)?

03 What are the roots of (x^2-19x+90=0)?

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

05 Which is the correct factorised form of the equation \(3x^2-11x+6=0\)?

06 What are the roots of the equation \\(3x^2-11x+6=0\\)?

07 Which values of \\(x\\) are obtained by solving \\(x-4)^2=25\\)?

08 While solving \(x^2+24x+17=0\) by the completing-square method, which number is added to make \(x^2+24x\) a perfect square?

09 Which of the following is equal to x² + 24x + 144?

10 What are the correct factors of (x^2+6x-55=0)?

11 What are the roots of (x^2+6x-55=0)?

12 What is the correct simplified form of the equation \(12x^2=108\) for solving it?

13 What are the solutions of the equation \(12x^2=108\)?

14 What is the correct part inside the square root in the quadratic formula?

15 If \(x^2+10x+k\) is a perfect square and can be written as \((x+5)^2\), what is the value of \(k\)?

16 In which method is the equation first changed into a form like ((x+p)^2=q)?

17 What is the correct factorised form of (25x^2-1=0)?

18 What are the roots of 5x² + 6x + 1 = 0?

19 In which perfect-square form can 16x² − 24x + 9 = 0 be written?

20 If 5x² + 9x = 2, what is the equation in standard form?

21 In which perfect-square form can 36x² − 60x + 25 = 0 be written?

22 On solving (x − 7)² = 11, what are the possible values of x?

23 If ((x-3)(x-7)=0), what are the solutions by zero product rule?

24 If (x-4)(x-9)=0, what are the solutions by the zero-product rule?

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