Q: Which of the following equations is a quadratic equation?

$x^2+3x+2=0$ A quadratic equation has the highest power (degree) of the variable $x$ equal to 2. In option A the highest power of $x$ is 2 (term $x^2$), so it is a quadratic equation. Option B is degree 1 (linear), option C is degree 3 (cubic), and option D has no variable so it is not an equation in $x$. Exam tip: To identify degree quickly, look for the term with the largest exponent of the variable and check that its coefficient is non-zero.

Q: What is the value of $a$ in the equation $5x^2-2x+9=0$?

5 In the standard form $ax^2+bx+c=0$, $a$ is the coefficient of $x^2$. In the given equation the coefficient of $x^2$ is 5, so $a=5$. Option B ($-2$) is the coefficient of $x$ (i.e. $b$), option C (9) is the constant term $c$, and option D (0) is not a coefficient present here. Exam tip: identify the number directly in front of $x^2$ to get $a$ quickly.

Q: What is the value of $b$ in $3x^2+7x-4=0$?

7 In the standard form $ax^2+bx+c=0$, $b$ is the coefficient of $x$. In $3x^2+7x-4=0$ the coefficient of $x$ is 7, so $b=7$. Common confusion: 3 is the coefficient of $x^2$ ($a$) and -4 is the constant term ($c$), so those choices are incorrect. Exam tip: Always rewrite or read the equation in standard form and directly pick $a, b, c$ before proceeding.

Q: In the equation $4x^2-8x+1=0$, what is the value of $c$?

1 A quadratic is written as $ax^2+bx+c=0$, where $c$ is the constant term. In $4x^2-8x+1=0$ the constant term is $1$, so $c=1$. The closest distractors: $-8$ is actually $b$, and $4$ is $a$; $-1$ has the wrong sign. Exam tip: Always rewrite the equation in standard form $ax^2+bx+c=0$ and then read off $a, b, c$.

Q: In the quadratic equation $x^2-6x=0$, what is the value of the constant term (c)?

0 Write the quadratic in standard form $ax^2+bx+c=0$. The given equation $x^2-6x=0$ can be written as $x^2-6x+0=0$, so the constant term is $c=0$. Option B ($-6$) is the coefficient $b$ of $x$, not the constant; options A and D are not present in the equation. Exam tip: always convert to $ax^2+bx+c=0$ and read off $a, b, c$ directly.

Q: What is the value of $x$ in the equation $9x^2 + 11 = 0$?

$\pm \dfrac{i\sqrt{11}}{3}$ Solve: $9x^2+11=0\Rightarrow 9x^2=-11\Rightarrow x^2=-\dfrac{11}{9}$. Taking square roots gives complex roots $x=\pm\sqrt{-\dfrac{11}{9}}=\pm\dfrac{i\sqrt{11}}{3}$. Option C is incorrect because it gives real roots $\pm\dfrac{\sqrt{11}}{3}$, which do not satisfy the equation; option D is wrong because $x=0$ yields 11, not 0. Quick exam tip: check the sign of the discriminant or move the constant to the RHS — a negative value under the square root indicates imaginary (complex) roots involving $i$.

Q: Under which condition is the equation $ax^2+bx+c=0$ called a quadratic equation?

$a\neq 0$ By definition a quadratic equation has degree 2, so the coefficient of $x^2$ must be nonzero, i.e. $a\neq 0$. If $a=0$ the equation reduces to the linear form $bx+c=0$. Options C and D ($b=0$ or $c=0$) are only special cases but do not prevent the equation from being quadratic as long as $a\neq 0$. Exam tip: always check the coefficient of $x^2$ to decide if an equation is quadratic.

Q: What is the standard form of the equation $2x^2+5x=3$?

$2x^2+5x-3=0$ Standard form means writing the quadratic as $ax^2+bx+c=0$ with all terms on one side. Moving the 3 from the right to the left changes its sign to −3, giving $2x^2+5x-3=0$. Option A is wrong because it keeps the constant as +3 instead of −3. Options C and D are wrong because they incorrectly change the sign of the linear term 5x. Exam tip: When converting to standard form, move all terms to one side and remember to change the sign of any term you transpose; then compare with $ax^2+bx+c=0$.

Q: What is the standard form of the equation $x^2 = 4x + 12$?

$x^2 - 4x - 12 = 0$ The standard form places all terms on one side and zero on the other. Moving the terms from the right to the left changes their signs, giving $x^2 - 4x - 12 = 0$. The other options fail because their signs are not changed correctly — e.g. option C has the constant sign wrong, options B and D have the linear term sign wrong. Exam tip: Bring all terms to one side and arrange in descending powers of $x$ ($x^2$, then $x$, then constant) before finalizing the answer.

Q: Which of the following equations is written in the standard form $ax^2+bx+c=0$?

x^2+2x+1=0 The standard form $ax^2+bx+c=0$ requires the right-hand side to be 0 and the coefficient $a$ of $x^2$ to be nonzero. Option A is already in that form (a=1, b=2, c=1). Options B and C have nonzero terms on the right as written (they can be rearranged into standard form, e.g. B → $x^2+2x-1=0$, but the question asks which is in standard form as given). Option D lacks an $x^2$ term so it is not a quadratic equation. Exam tip: verify RHS = 0 and that the $x^2$ coefficient is not zero to confirm standard form quickly.

Question 1

Which of the following equations is a quadratic equation?

Accepted Answer

$x^2+3x+2=0$ A quadratic equation has the highest power (degree) of the variable $x$ equal to 2. In option A the highest power of $x$ is 2 (term $x^2$), so it is a quadratic equation. Option B is degree 1 (linear), option C is degree 3 (cubic), and option D has no variable so it is not an equation in $x$. Exam tip: To identify degree quickly, look for the term with the largest exponent of the variable and check that its coefficient is non-zero.

Question 2

What is the value of $a$ in the equation $5x^2-2x+9=0$?

Accepted Answer

5 In the standard form $ax^2+bx+c=0$, $a$ is the coefficient of $x^2$. In the given equation the coefficient of $x^2$ is 5, so $a=5$. Option B ($-2$) is the coefficient of $x$ (i.e. $b$), option C (9) is the constant term $c$, and option D (0) is not a coefficient present here. Exam tip: identify the number directly in front of $x^2$ to get $a$ quickly.

Question 3

What is the value of $b$ in $3x^2+7x-4=0$?

Accepted Answer

7 In the standard form $ax^2+bx+c=0$, $b$ is the coefficient of $x$. In $3x^2+7x-4=0$ the coefficient of $x$ is 7, so $b=7$. Common confusion: 3 is the coefficient of $x^2$ ($a$) and -4 is the constant term ($c$), so those choices are incorrect. Exam tip: Always rewrite or read the equation in standard form and directly pick $a, b, c$ before proceeding.

Question 4

In the equation $4x^2-8x+1=0$, what is the value of $c$?

Accepted Answer

1 A quadratic is written as $ax^2+bx+c=0$, where $c$ is the constant term. In $4x^2-8x+1=0$ the constant term is $1$, so $c=1$. The closest distractors: $-8$ is actually $b$, and $4$ is $a$; $-1$ has the wrong sign. Exam tip: Always rewrite the equation in standard form $ax^2+bx+c=0$ and then read off $a, b, c$.

Question 5

In the quadratic equation $x^2-6x=0$, what is the value of the constant term (c)?

Accepted Answer

0 Write the quadratic in standard form $ax^2+bx+c=0$. The given equation $x^2-6x=0$ can be written as $x^2-6x+0=0$, so the constant term is $c=0$. Option B ($-6$) is the coefficient $b$ of $x$, not the constant; options A and D are not present in the equation. Exam tip: always convert to $ax^2+bx+c=0$ and read off $a, b, c$ directly.

Question 6

What is the value of $x$ in the equation $9x^2 + 11 = 0$?

Accepted Answer

$\pm \dfrac{i\sqrt{11}}{3}$ Solve: $9x^2+11=0\Rightarrow 9x^2=-11\Rightarrow x^2=-\dfrac{11}{9}$. Taking square roots gives complex roots $x=\pm\sqrt{-\dfrac{11}{9}}=\pm\dfrac{i\sqrt{11}}{3}$. Option C is incorrect because it gives real roots $\pm\dfrac{\sqrt{11}}{3}$, which do not satisfy the equation; option D is wrong because $x=0$ yields 11, not 0. Quick exam tip: check the sign of the discriminant or move the constant to the RHS — a negative value under the square root indicates imaginary (complex) roots involving $i$.

Question 7

Under which condition is the equation $ax^2+bx+c=0$ called a quadratic equation?

Accepted Answer

$a\neq 0$ By definition a quadratic equation has degree 2, so the coefficient of $x^2$ must be nonzero, i.e. $a\neq 0$. If $a=0$ the equation reduces to the linear form $bx+c=0$. Options C and D ($b=0$ or $c=0$) are only special cases but do not prevent the equation from being quadratic as long as $a\neq 0$. Exam tip: always check the coefficient of $x^2$ to decide if an equation is quadratic.

Question 8

What is the standard form of the equation $2x^2+5x=3$?

Accepted Answer

$2x^2+5x-3=0$ Standard form means writing the quadratic as $ax^2+bx+c=0$ with all terms on one side. Moving the 3 from the right to the left changes its sign to −3, giving $2x^2+5x-3=0$. Option A is wrong because it keeps the constant as +3 instead of −3. Options C and D are wrong because they incorrectly change the sign of the linear term 5x. Exam tip: When converting to standard form, move all terms to one side and remember to change the sign of any term you transpose; then compare with $ax^2+bx+c=0$.

Question 9

What is the standard form of the equation $x^2 = 4x + 12$?

Accepted Answer

$x^2 - 4x - 12 = 0$ The standard form places all terms on one side and zero on the other. Moving the terms from the right to the left changes their signs, giving $x^2 - 4x - 12 = 0$. The other options fail because their signs are not changed correctly — e.g. option C has the constant sign wrong, options B and D have the linear term sign wrong. Exam tip: Bring all terms to one side and arrange in descending powers of $x$ ($x^2$, then $x$, then constant) before finalizing the answer.

Question 10

Which of the following equations is written in the standard form $ax^2+bx+c=0$?

Accepted Answer

x^2+2x+1=0 The standard form $ax^2+bx+c=0$ requires the right-hand side to be 0 and the coefficient $a$ of $x^2$ to be nonzero. Option A is already in that form (a=1, b=2, c=1). Options B and C have nonzero terms on the right as written (they can be rearranged into standard form, e.g. B → $x^2+2x-1=0$, but the question asks which is in standard form as given). Option D lacks an $x^2$ term so it is not a quadratic equation. Exam tip: verify RHS = 0 and that the $x^2$ coefficient is not zero to confirm standard form quickly.

Question 11

If $x=2$, what is the value of the left-hand side of the equation $x^2-5x+6=0$?

Accepted Answer

0 Substitute and evaluate: $2^2-5\cdot2+6=4-10+6=0$. Hence the left-hand side equals 0 and $x=2$ is a root of the quadratic. Option C (4) is incorrect — that would result from using only $2^2$ and ignoring the other terms. Exam tip: compute powers and multiplications first, then perform additions/subtractions to avoid sign or order errors.

Question 12

Which value is one solution of $x^2-9=0$?

Accepted Answer

3 Check by substitution: $3^2-9=9-9=0$, so 3 is a root. Factoring gives $x^2-9=(x-3)(x+3)$, so the roots are $x=3$ and $x=-3$ (note -3 is not among the choices). The nearest distractor 2 fails because $2^2-9=4-9=-5\neq0$. Exam tip: for simple quadratics either factor or test small integer values to verify roots quickly.

Question 13

What do we get when we substitute $x=2$ into the equation $x^2-4x+4=0$?

Accepted Answer

$0=0$ Substituting $x=2$ gives $2^2-4\cdot2+4=4-8+4=0$. So the equation becomes $0=0$, which means $x=2$ is a correct solution (root). Also note the polynomial is $(x-2)^2$, so this is a repeated root (discriminant zero). Exam tip: always compute each term carefully when substituting; recognizing a perfect square trinomial saves time.

Question 14

Which is the standard form of a quadratic equation, where $a\ne 0$?

Accepted Answer

$ax^2+bx+c=0$ A quadratic equation has highest power 2, so its standard form is $ax^2+bx+c=0$, with $a\ne0$. Option B is linear because its highest power is 1. Exam tip: identify the equation type by checking the highest exponent.

Question 15

Which of the following equations is not a quadratic equation?

Accepted Answer

$x^3 + x + 1 = 0$ The degree of a polynomial is the highest exponent of the variable; a quadratic equation must have degree 2. Option C has highest exponent $3$ ($x^3+x+1=0$), so it is not quadratic. The other options are degree 2: A is degree $2$, B is degree $2$, and D becomes $4x^2-7=0$ which is also degree $2$. The closest distractor is D because it is written as $4x^2=7$ and may look different, but converting to standard form shows it is quadratic. Exam tip: convert to standard form $ax^2+bx+c=0$ and check the highest power of the variable.

Question 16

For the equation $6x^2=0$, what are a, b and c respectively?

Accepted Answer

(6,0,0) Write the quadratic in standard form $ax^2+bx+c=0$. The equation $6x^2=0$ is equivalent to $6x^2+0x+0=0$, so $a=6,\; b=0,\; c=0$. Option C is wrong because it incorrectly gives the x-coefficient as 1 instead of 0; B and D place numbers in wrong positions. Exam tip: explicitly add missing terms with coefficient 0 and then read off a, b, c from the standard form.

Question 17

Write the expression $x(x+5)=0$ in the standard quadratic form $ax^2+bx+c=0$.

Accepted Answer

x^2+5x=0 Expanding gives \x(x+5)=x\cdot x + x\cdot 5 = x^2+5x\. In standard form this is \x^2+5x+0=0\ or simply \x^2+5x=0\. Option B has the wrong sign (would be \-5x\ if the second term were negative), option C incorrectly multiplies the square term by 5 (\5x^2\ instead of \x^2\), and option D is linear, not quadratic. Exam tip: expand brackets carefully and collect terms on one side to match \ax^2+bx+c=0\ before identifying a, b, c.

Question 18

What is the expanded form of (x+2)(x+3)=0?

Accepted Answer

$x^2+5x+6=0$ Multiply the two binomials using FOIL: $(x+2)(x+3)=x\cdot x + x\cdot 3 + 2\cdot x + 2\cdot 3 = x^2+3x+2x+6$. Combining like terms gives $x^2+5x+6=0$, so A is correct. The closest distractor B ($x^2+6x+5=0$) simply swaps coefficients and constant but does not follow from multiplication; D has the wrong sign on the constant. Exam tip: use FOIL (or multiply each term) and then combine like terms to avoid sign or order mistakes.

Question 19

What is the standard form of $(x-4)(x+1)=0$?

Accepted Answer

$x^2-3x-4=0$ Expanding gives $(x-4)(x+1)=x^2+x-4x-4=x^2-3x-4$, so the standard form is $x^2-3x-4=0$. Quick check: the roots are 4 and −1, their sum is 3, and since −b/a = 3 we must have b = −3. The closest distractor (option B) typically arises from a sign mistake on the linear term; the other options have wrong signs or constant term. Exam tip: combine like terms immediately after multiplying and verify using sum/product of roots.

Question 20

What is the coefficient of (x^2) in (x^2+8x+16=0)?

Accepted Answer

(1) Direct answer: Option B, 1. A quadratic equation is commonly written as $ax^2+bx+c=0$, where a is the coefficient of $x^2$. In the expression $x^2+8x+16=0$, the first term can be written as $1x^2$. Therefore a=1. Option A, 0, is wrong because the term is present, not absent. Option B is correct because an unwritten numerical factor is understood to be one. Option C, 8, is the coefficient of x, not of $x^2$. Option D, 16, is the constant term because it has no x. Read the power carefully: the coefficient belongs to the term with that exact power. Exam cue: an alone-standing variable term has coefficient 1.

Question 21

In the equation $-x^2+3x+10=0$, what is the value of $a$?

Accepted Answer

-1 In the standard form $ax^2+bx+c=0$, $a$ is the coefficient of $x^2$. In $-x^2+3x+10=0$ the $x^2$ term is $-x^2$, so its coefficient is $-1$. Therefore $a=-1$. Note: 3 is the coefficient $b$ of $x$, 10 is the constant $c$, and 1 is the wrong sign. Exam tip: rewrite any quadratic as $ax^2+bx+c=0$ and compare coefficients term by term.

Question 22

Which of the following expresses the equation $x^2+6x+9=0$ in factorised form?

Accepted Answer

$(x+3)^2=0$ $x^2+6x+9$ is a perfect-square trinomial. Comparing with $(x+a)^2=x^2+2ax+a^2$ gives $2a=6$, so $a=3$ and the trinomial equals $(x+3)^2$. Hence the equation factors as $(x+3)^2=0$. Option B is incorrect because $(x-3)^2$ would produce a middle term of $-6x$; options C and D have wrong constant terms. Exam tip: identify perfect squares by checking if the constant equals the square of half the middle coefficient (i.e., check if constant = (middle/2)^2).

Question 23

Which of the following values is a solution of the equation $x^2 + x - 6 = 0$?

Accepted Answer

2 Factor the quadratic: $x^2+x-6=(x+3)(x-2)$. Thus the roots are $x=-3$ and $x=2$. The value 2 is among the options, so it is the correct answer. Checking the nearest distractor 1 gives $1^2+1-6=-4$, not zero, so it is incorrect. Exam tip: either factor the quadratic or substitute the option values to verify quickly.

Question 24

If an equation in x has degree 2, what is it called?

Accepted Answer

Quadratic equation The governing definition is the degree of a polynomial equation. The degree is the greatest exponent of the variable that has a non-zero coefficient. When the highest power of x is 2, the equation is called a quadratic equation. Its usual standard form is ax² + bx + c = 0, where a is not zero; this condition ensures that the x² term is actually present. A linear equation has degree 1, a cubic equation has degree 3, and a constant equation has degree 0 because it contains no variable term of positive degree. Therefore, an equation of degree 2 is precisely a quadratic equation, making option B correct. The other choices refer to different degrees.

Question 25

Which of the following contains the $x^2$ term?

Accepted Answer

$x^2-2=0$ A quadratic (second-degree) expression must include a nonzero $x^2$ term. Option C, $x^2-2=0$, clearly contains the $x^2$ term so it is correct. The other options, $5x+2=0$, $7x-9=0$, and $3x+4=7$, are linear equations whose highest power of $x$ is 1 and therefore do not contain $x^2$. Exam tip: To identify a quadratic quickly, check the highest power of the variable — if it is 2, the equation is quadratic.

Introduction to Quadratic Equations

Practice questions

01 Which of the following equations is a quadratic equation?

02 What is the value of \(a\) in the equation \(5x^2-2x+9=0\)?

03 What is the value of \(b\) in \(3x^2+7x-4=0\)?

04 In the equation \(4x^2-8x+1=0\), what is the value of \(c\)?

05 In the quadratic equation \(x^2-6x=0\), what is the value of the constant term (c)?

06 What is the value of \(x\) in the equation \(9x^2 + 11 = 0\)?

07 Under which condition is the equation \(ax^2+bx+c=0\) called a quadratic equation?

08 What is the standard form of the equation \(2x^2+5x=3\)?

09 What is the standard form of the equation \(x^2 = 4x + 12\)?

10 Which of the following equations is written in the standard form \(ax^2+bx+c=0\)?

11 If \(x=2\), what is the value of the left-hand side of the equation \(x^2-5x+6=0\)?

12 Which value is one solution of \(x^2-9=0\)?

13 What do we get when we substitute \(x=2\) into the equation \(x^2-4x+4=0\)?

14 Which is the standard form of a quadratic equation, where \(a\ne 0\)?

15 Which of the following equations is not a quadratic equation?

16 For the equation \(6x^2=0\), what are a, b and c respectively?

17 Write the expression \\(x(x+5)=0\\) in the standard quadratic form \\(ax^2+bx+c=0\\).

18 What is the expanded form of (x+2)(x+3)=0?

19 What is the standard form of \((x-4)(x+1)=0\)?

20 What is the coefficient of (x^2) in (x^2+8x+16=0)?

21 In the equation \(-x^2+3x+10=0\), what is the value of \(a\)?

22 Which of the following expresses the equation \(x^2+6x+9=0\) in factorised form?

23 Which of the following values is a solution of the equation \(x^2 + x - 6 = 0\)?

24 If an equation in x has degree 2, what is it called?

25 Which of the following contains the \(x^2\) term?

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